Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.46.0-wmf.24 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 4 28 2806681 2806611 2026-04-26T15:39:09Z Codename Noreste 2969951 /* Add some user rights to the curator user group? */ reply: I went ahead and filed phab:T424445. (-) ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2806681 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Requested update to [[Wikiversity:Interface administrators]] == Currently, [[Wikiversity:Interface administrators]] is a policy that includes a caveat that interface admins are not required long-term and that user right can only be added for a period of up to two weeks. I am proposing that we remove this qualification and allow for indefinite interface admin status. I think this is useful because there are reasons for tweaking the site CSS or JavaScript (e.g. to comply with dark mode), add gadgets (e.g. importing Cat-a-Lot, which I would like to do), or otherwise modifying the site that could plausibly come up on an irregular basis and requiring the overhead of a bureaucrat to add the user rights is inefficient. In particular, I am also going to request this right if the community accepts indefinite interface admins. Thoughts? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:23, 17 August 2025 (UTC) :And who will then monitor them to make sure they don't damage the project in any way, or abuse the rights acquired in this way? For large projects, this might not be a problem, but for smaller projects like the English Wikiversity, I'm not sure if there are enough users who would say, something is happening here that shouldn't be happening. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:28, 20 August 2025 (UTC) ::Anyone would be who. This argument applies to any person with any advanced rights here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:46, 20 August 2025 (UTC) :I think it is reasonable to allow for longer periods of access than 2 weeks to interface admin and support adjusting the policy to allow for this flexibility. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:57, 2 December 2025 (UTC) ::+1 —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:38, 25 January 2026 (UTC) :@[[User:Koavf|Koavf]] I agree that the two-week requirement could be revised, but wouldn’t people just request access for a specific purpose anyway? Instead of granting indefinite access, they should request the specific time frame they need the rights for—until the planned fixes are completed—and then request an extension if more time is required. We could remove the two-week criterion while still keeping the access explicitly temporary. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:48, 25 January 2026 (UTC) ::I just don't see why this wiki needs to be different than all of the others. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 25 January 2026 (UTC) :::There isn’t really much of a need for a permanent one at this point in time [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 25 January 2026 (UTC) :I quite agree with this proposal, so long as they perform the suggested changes as mentioned here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 04:06, 26 January 2026 (UTC) :: Just to clarify, I support '''indefinite interface admin status'''. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:34, 13 April 2026 (UTC) :I think there is decent consensus for lengthening this, but not necessarily for indefinite permissions, so does anyone object to me revising it to the standard being 120 days instead of two weeks? I'll check back on this thread in three weeks and if there's no objection, I'll make the change. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:47, 13 April 2026 (UTC) ::Sure [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:27, 13 April 2026 (UTC) ::Thanks for proposing this, Justin. I agree with the proposal to lengthen the interface admin period from 2 weeks but not indefinitely. Can I check the source(s) for the standard being 120 days (I'm guessing policies on other projects or maybe global policy?)? In any case, I think it is reasonable for us to adopt a similar period. However, note on the current policy discussion page notes from @[[User:Dave Braunschweig|Dave Braunschweig]] arguing for shorter periods to lower risk, that's why it is 2 weeks. But if there are projects that need longer access, that should also be accommodated. Maybe we could adjust the policy to specify that ''interface admin rights can be given for 14 to 120 days depending on how long is required and what is supported by the community''. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:29, 24 April 2026 (UTC) :::There was there was no source for 120: it was just more than 14 and less than infinity. The "14 to 120" also seems reasonable. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 14:33, 24 April 2026 (UTC) ::: On some small/medium-sized wikis, such as English Wikibooks and English Wikiquote for example, indefinite interface administrator access for administrators is allowed, but they tend not to make changes to the CSS and JS page changes unless it's truly necessary. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:34, 24 April 2026 (UTC) :::It's a good idea to make the length of this right on request or allow to be prolonged. However, IA should test large changes somewhere else, for example on the en.wv mirror, and only after testing it on the mirror, adapt it to the live version. That means I can't imagine a time-consuming operation right now. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:04, 24 April 2026 (UTC) ::::Sorry, what mirror is this? Are you talking about beta.wv? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:32, 24 April 2026 (UTC) :::::Not beta.wv. Basically somewhere else then on a live wiki. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:59, 24 April 2026 (UTC) :::::: Wouldn't testing on a user's own common.css page work anyway? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:36, 24 April 2026 (UTC) == [[Wikiversity:Curators|Curators and curators policy]] == How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC) :It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|<span style="font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;">RailwayEnthusiast2025</span>]] ^{[[User talk:RailwayEnthusiast2025|<span style="color:#59a53f">''talk with me!''</span>]]} 18:33, 27 October 2025 (UTC) :I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:08, 3 December 2025 (UTC) :What? I thought you were getting it approved, Juandev... :) [[User:I&#39;m Mr. Chris|I&#39;m Mr. Chris]] ([[User talk:I&#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC) ::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC) :::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC) ::::Agree -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:00, 27 March 2026 (UTC) Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:40, 24 April 2026 (UTC) == [[Template:AI-generated]] == After going through the plethora of ChatGPT-generated pages made by [[User:Lbeaumont|Lbeaumont]] (with many more pages to go), I'd like community input on this proposal to [[Wikiversity:Artificial intelligence]] that I think would be benefical for the community: *Resources generated by AI '''must''' be indicated as so through the project box, [[Template:AI-generated]], on either the page or the main resource (if the page is a part of a project). I do not believe including a small note/reference that a page is AI-generated is sufficient, and I take my thinking from [[WV:Original research|Wikiversity's OR policy]] for OR work: ''Within Wikiversity, all original research should be clearly identified as such''. I believe resources created from AI should also be clearly indicated as such, especially since we are working on whether or not AI-generated resources should be allowed on the website (discussion is [[Wikiversity talk:Artificial intelligence|here]], for reference). This makes it easier for organizational purposes, and in the event ''if'' we ban AI-generated work. I've left a message on Lee's talk page over a week ago and did not get a response or acknowledgement, so I'd like for the community's input for this inclusion to the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 26 January 2026 (UTC) :I believe that existing Wikiversity policies are sufficient. Authors are responsible for the accuracy and usefulness of the content that is published. This policy covers AI-generated content that is: 1) carefully reviewed by the author publishing it, and 2) the source is noted.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:38, 27 January 2026 (UTC) ::A small reference for pages that are substantially filled with Chat-GPT entries, like [[Real Good Religion]], [[Attributing Blame]], [[Fostering Curiosity]], are not sufficient IMO and a project box would be the best indicator that a page is AI-generated (especially when there is a mixture of human created content AND AI-generated content, as present in a lot of your pages). This is useful, especially considering the notable issues with AI (including hallucinations and fabrication of details), so viewers and support staff are aware. These small notes left on the pages are not as easily viewable as a project box or banner would be. I really don't see the issue with a clear-label guideline. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:34, 27 January 2026 (UTC) ::{{ping|Lbeaumont}} I noticed your reversions [https://en.wikiversity.org/w/index.php?title=Exploring_Existential_Concerns&diff=prev&oldid=2788278 here] & [https://en.wikiversity.org/w/index.php?title=Subjective_Awareness&diff=prev&oldid=2788257 here]. I'd prefer to have a clean conversation regarding this proposition. Please voice your concerns here. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 28 January 2026 (UTC) :::Regarding Subjective Awareness, I distinctly recall the effort I went to to write that the old-fashioned way. It is true that ChatGPT assisted me in augmenting the list of words suggested as candidate subjective states. This is a small section of the course, is clearly marked, and makes no factual claim. Marking the entire course as AI-generated is misleading. I would have made these comments when I reverted your edit; however, the revert button does not provide that opportunity. :::Regarding the Exploring Existential Concerns course, please note this was adapted from my EmotionalCompetency.com website, which predates the availability of LLMs. The course does include two links, clearly labeled as ChatGPT-generated. Again, marking the entire course as AI-generated is misleading. :::On a broader issue, I don't consider your opinions to have established a carefully debated and adopted Wikiversity policy. You went ahead and modified many of my courses over my clearly stated objections. Please let this issue play out more completely before editing my courses further. Thanks.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:11, 29 January 2026 (UTC) ::::Understood, and I respect your position. I apologize if my edits were seen as overarching. We could change the project box to "a portion of this resource was generated by AI", or something along those lines. Feel free to revert my changes where you see fit, and I encourage more users to provide their input. EDIT: I've made changes to the template to indicate that a portion of the content has been generated from an LLM. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:50, 29 January 2026 (UTC) :::::Thanks for this reply. The new banner is unduly large and alarming. There is no need for alarm here. The use of AI is not harmful per se. Like any technology, it can be used to help or to harm. I take care to craft prompts carefully, point the LMM to reliable source materials, and to carefully read and verify the generated text before I publish it. This is all in keeping with long-established Wikiversity policy. We don't want to use a  [[w:One-drop_rule|one-drop rule]] here or cause a [[w:Satanic_panic|satanic panic]]. We can learn our lessons from history here. I don't see any pedagogical reason for establishing a classification of "AI generated", but if there is a consensus that it is needed, perhaps it can be handled as just another category that learning resources can be assigned to. I would rather focus on identifying any errors in factual claims than on casting pejorative bias toward AI-generated content. An essay on the best practices for using LMM on Wikiveristy would be welcome.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:58, 30 January 2026 (UTC) ::::::The new banner mimics the banner that is available on the English Wikibooks (see [[b:Template:AI-generated]] & [[b:Template:Uses AI]]), so my revisions aren't unique in this aspect. At this point, I'd welcome other peoples' inputs. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:40, 30 January 2026 (UTC) == How do I start making pages? == Is there a notability guideline for Wikiversity? What is the sourcing policy for information? What is the Manual of Style? What kind of educational content qualifies for Wikiversity? All the introduction pages are a bit unclear. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 02:25, 28 January 2026 (UTC) :{{ping|VidanaliK}} Welcome to Wikiversity! I've left you a welcome message on your talk page. That should help you out. Make sure to especially look at [[Wikiversity:Introduction]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:11, 28 January 2026 (UTC) ::It says that I can't post more pages because I have apparently exceeded the new page limit. How long does it take before that new page limit expires? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 16:57, 28 January 2026 (UTC) :::This is a restriction for new users so that Wikiversity is not hit with massive spam. As for when this limit will expire, it should be a few days or after a certain number of edits. It's easy to overcome, though I do not have the exact numbers atm. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:08, 29 January 2026 (UTC) ::::OK, I think I got past the limit. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 17:21, 29 January 2026 (UTC) ==Why does it feel like Wikiversity is no longer really active anymore?== I've been looking at recent changes, and both today and yesterday there haven't been many changes that I haven't made; it feels like walking through a ghost town, is this just me or is Wikiversity not really active anymore? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 03:54, 30 January 2026 (UTC) :There is fewer people editing these days compared to the past. Many newcomers tend to edit in Wikipedia instead. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 06:39, 30 January 2026 (UTC) :It’s a little slow, but I’m happy to know that Wikiversity is a place that I think should provide value even if the activity of editors fluctuates. If it’s any consolation your edits may be encouraging for some anonymous newcomer to start edits on their own! I think it’s hard to build community when there is such a wide variety of interests and a smaller starting userbase. Also sometimes the getting into a particular topic that already exists can be intimidating because some relics (large portals, school, categories, etc.) have intricate, unique and generally messy levels of organization. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:16, 9 March 2026 (UTC) == Inactivity policy for Curators == I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC) :Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC) ::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC) :::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC) ::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC) ::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]: ::::::The maximum time period of inactivity <u>without community review</u> for curators is two years (consistent with the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights. ::::: -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:51, 27 March 2026 (UTC) :::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC) :::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC) :I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:59, 24 April 2026 (UTC) :: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC) == [[Wikiversity:Artificial intelligence]] to become an official policy == {{Archive top|After running for a week, there is consensus, alongside comments, for [[Wikiversity:Artificial intelligence]] to be implemented as an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:27, 17 April 2026 (UTC)}} With the introduction of AI-material, and some material just plain disruptive, its imperative that Wikiversity catches up with its sister projects and implements an official AI policy that we can work with. The recent issue of [[User:Lbeaumont|Lbeaumont]]'s 50+ articles that contain significantly large AI-generated material has made me came to the Colloquium. This user has also been removing the [[Template:AI-generated]] template from their pages, calling it "misleading", "alarmist", and "pejorative" - which is all just simply nonsensical rationales. Not to even mention this user's contributions to the English Wikipedia have been [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Inner_Development_Goals contested] and [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Multipolar_trap removed] a couple of times (for being low-quality and clearly LLM-generated), highlighting the need for an actual policy to be implemented here on Wikiversity. I would like to ping {{ping|Juandev}} and {{ping|Jtneill}} for their thoughts as well, since I'd like this to be implemented as soon as possible. Wikiversity has a significant issue with implementing anti-disruptive measures, hence why we have received numerous complaints as a community about our quality. I originally was reverting the removal of the templates, but realized that this is still a proposed policy, which it shouldn't be anymore. It should be a recognized Wikiversity policy. 14:54, 10 March 2026 (UTC) —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:54, 10 March 2026 (UTC) :@[[User:Atcovi|Atcovi]] '''I agree''' that the draft, should become official policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:00, 10 March 2026 (UTC) :I provided a detailed response at: [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI]] :I will appreaciate it if you consder that carefully. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 22:49, 10 March 2026 (UTC) :Agree it should become official Wikiversity policy on the condition <u>that point point 5 is about [significant/substantial] LLM-generated text specifically</u>. Not a good idea to overuse it, it should be added when there is substantial AI-generated text on the page, not for other cases. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:37, 11 March 2026 (UTC) :What policy is being debated? Is it the text on this page, which is pointed to by the general banner, or the text at:   [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence,]]   which is pointed to by the specific banner? Let's begin with coherence on the text being debated. Thanks! [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:49, 17 March 2026 (UTC) ::@[[User:Lbeaumont|Lbeaumont]] This is a call for approval of the new Wikiversity policy. You expressed your opinion [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI|on the talk page of the proposal]], I replied to you and await your response.When creating policies, it is necessary to propose specific solutions. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:12, 17 March 2026 (UTC) :::Toward a Justified and Parsimonious AI Policy :::As we collaborate to develop a consensus policy on the use of Large Language Models, it is wise to begin by considering the needs of the various stakeholders to the policy. :::The stakeholders are: :::1)     The users, :::2)     The source providers, and :::3)     The editors :::There may also be others with a minor stake in this policy, including the population at large. :::The many needs of the users are currently addressed by long-standing [[Wikiversity:Policies|Wikiversity policies]], so we can focus on what, if any, additional needs arise as LLMs are deployed. :::As always, users need assurance that propositional statements are accurate. This is covered by the existing policy on [[Wikiversity:Verifiability|verifiably]]. In addition, it is expected by both the users and those that provide materials used as sources for the text are [[Wikiversity:Cite sources|accurately attributed]]. This is also covered by [[Wikiversity:Cite sources|existing policies]]. :::To respect the time and effort of editors, a parsimonious policy will unburden editors from costly requirements that exceed benefits to the users. :::Finally, it is important to recognize that because attention is our most valuable seizing attention unnecessarily is a form of theft. :::The following proposed policy statement results from these considerations: :::Recommended Policy statement: :::·       Editors [[Wikiversity:Verifiability|verify the accuracy]] of propositional statements, regardless of the source. :::·       Editors [[Wikiversity:Cite sources|attribute the source]] of propositional statements. In the case of LLM, cite the LLM model and the prompt used. :::·       Use of various available templates to mark the use of LLM are optional. Templates that are flexible in noting the type and extend of LLM usage are preferred. Templates that avoid unduly distracting or alarming the user are preferred. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:56, 19 March 2026 (UTC) ::::Do we discuss here or there? I have replied you there as your proposal is about that policy so it is tradition to discuss it at the affected talk page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:59, 19 March 2026 (UTC) : {{support}} Thanks for the proposed policy development and discussion; also note proposed policy talk page discussion: [[Wikiversity talk:Artificial intelligence]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:05, 24 March 2026 (UTC) ::I think the Wikiversity AI policy shall be official. – [[User:RestoreAccess111|RestoreAccess111]] <sup style="font-family:Arimo, Arial;">[[User talk:RestoreAccess111|Talk!]]</sup> <sup style="font-family:Times New Roman, Tinos;">[[Special:Contributions/RestoreAccess111|Watch!]]</sup> 06:11, 13 April 2026 (UTC) {{archive bottom}} == New titles for user right nominations == <div class="cd-moveMark">''Moved from [[Wikiversity talk:Candidates for Custodianship#New titles for user right nominations]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 17 April 2026 (UTC)''</div> I would like to propose the following retitles should a user be nominated for any of the following user rights: * Curator: Candidates for Curatorship * Bureaucrat: Candidates for Bureaucratship The reason is that many curator (and probably bureaucrat) requests have run solely under {{tq|Candidates for Custodianship}}, but that title might sound misleading (especially in regards to the permission a user is requesting). CheckUser and Oversight (suppressor) are not included above since no user was nominated for these sensitive permissions, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:30, 19 March 2026 (UTC) :And it's not that when someone at the beginning misplaced the request, no one thought to move it and the others copied it. Even today, it would be possible to simply take it all and move it. Otherwise, for me, the more fundamental problem is that there is [[Wikiversity:Curators|no approved policy for curators]] than where the requests are based. Curators then operate in a certain vacuum and if one of them "breaks out of the chain", the average user doesn't have many transparent tools to deal with it, because there is no policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:02, 19 March 2026 (UTC) ::I am not talking about the curator page (policy proposal). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:08, 21 March 2026 (UTC) : @[[User:Juandev|Juandev]] I'll see if I can do an overhaul of [[Wikiversity:Candidates for Custodianship]], just like I recently did with the Requests for adminship page on English Wikiquote. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:17, 18 April 2026 (UTC) == Technical Request: Courtesy link.. == [[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC) : I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC) :: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :) [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC) :I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC) : I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC) == WikiEducator has closed == Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/]. It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating. They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki. The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license). The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 1 April 2026 (UTC) :I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:46, 1 April 2026 (UTC) == Wikinews is ending == Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]). And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC) :Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well. :In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up "math properties" for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). :I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC) :For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest "lagging behind" or "failure" project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:46, 2 April 2026 (UTC) :[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:03, 11 April 2026 (UTC) == Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) == Hello everyone, This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as <code>Module:Itemgroup/list</code>). '''The Change:'''<br /> Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''. '''What You Need To Do:'''<br /> To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search. '''Deadline:'''<br /> We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles. Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:11, 3 April 2026 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=29941252 --> :I didnt find such properties, so we are probably fine. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:00, 12 April 2026 (UTC) :: +1 (agreed). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:19, 12 April 2026 (UTC) == Enable the abuse filter block action? == In light of [[Special:AbuseLog/80178]] (coupon spam), I would like to propose enabling the block action for the abuse filter. Only custodians will be able to enable and disable that action on an abuse filter, and it is useful to block ongoing vandalism. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:12, 13 April 2026 (UTC) :Seems like a good idea, almost all of the users which create such pages are spambots so this shouldn’t be a problem. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:41, 13 April 2026 (UTC) :Can you explain some more (I am new to abuse filters)? It looks like the attempted edit was prevented? Which abuse filter? :Note on your suggestion, have also reactivated Antispam Filter 12 - see [[WV:RCA]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:45, 15 April 2026 (UTC) :: I am proposing that we activate the abuse filter block action, which if a user triggers an abuse filter, it would actually block the user in question - the same mechanism that a custodian would use to block users. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:11, 15 April 2026 (UTC) :::OK, thankyou, that makes sense. And, reviewing the abuse filter 12 log, it would be helpful because it would prevent the need for manual blocking. But I don't see a setting for autoblocking? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:14, 15 April 2026 (UTC) :::: I think it probably adds an autoblock. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:43, 16 April 2026 (UTC) : [[User:Jtneill|Jtneill]] and [[User:PieWriter|PieWriter]], given that a little bit more than a week has passed and there is minimal consensus to activate the abuse filter block action, I filed [[phab:T424053]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:05, 21 April 2026 (UTC) ::Thank-you for doing this. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:03, 24 April 2026 (UTC) == Advice needed: A Neurodiversity-inspired Idea/observation == If I want the greatest participation of others to "provide constructive criticism to my idea" or to "shoot down my idea" or "idea". What I've called it so far is "The Neurodiversity-inspired Idea". At other times I used more sensationalist wording but here on Wikiversity I don't dare do that. I actually woke up with thinking about putting this into my userspace draft: "Personal Observations Made By Meeting Autistic and Non-Autistic Adults". My ultimate goal is to stop blathering about my "idea" to friend and family without feeling my "methodology" is going into any progressive direction whatsoever. My latest encounter was somewhat constructive though. A friend of a friend who worked with people presenting ideas in attempting to getting grants. I don't want a grant. I just want to figure out how I can express my "idea" in a way so that I can more clearly figure out what flaws it got. At the same time I tend to overthink. If anyone thinks etherpad might be a good place and considering Wikimedia already got an etherpad at https://etherpad.wikimedia.org/ if anyone feels like they know me better in the future feel free to suggest a "session" on etherpad. '''If I don't receive a reply to this in 1 week's time I will begin to explore this "idea" into my userspace''' unless you replied and refrained me from doing so, of course. Then maybe after "developing it there" I might reference it to you another future time here in the Colloquium, with my "idea" still in my userspace draft. This "idea" is sort of a burden, I'm happy I've made the choice to get rid of it and hopefully move on with my life, unless there is something to this "idea". My failure is probably evident: I feel I haven't told you anything. Same happened to when I talked to friends and family. In danger of overthinking it further I'll publish this right now. I need to "keep it together" [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:36, 16 April 2026 (UTC) :Good on you putting it out there ... and hitting publish :). I'd say go for it (no need to wait), give birth to your idea and share about it here and elsewhere. Let it take shape and see where it might go. In many ways, this is exactly what an open collaborative learning community should be doing. Others might not know well how to respond, so perhaps consider creating some questions to accompany the idea. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:21, 16 April 2026 (UTC) ::Thank you for encouraging me in developing the idea. ::I have created a "questions" section in the draft which is visible in the table of contents now. My brain was "frozen" today metaphorically speaking in that I felt I had like a "writer's block" so the draft has more "AI/LLM" content than before. I used the LLM for generating questions. The answers are so far human-only. ::I've also created a subsection where I could add the prompts that made the LLM generate the questions. That could help people make better prompts perhaps. I've described what it is about inside of it and there are some chaotically written notes. ::[[Draft:The_Neurodiversity-inspired_Idea#Questions_that_might_encourage_the_development_of_this_idea_and_its_methodology]] ::My draft is missing stuff. Any questions that you contribute to my draft will probably help me and if I don't understand the questions I'll probably notify you and also at the same time "feed them" to an LLM and ask in my input like "explain in simple words what this question means, what is it searching for?" etc. while I wait for an answer. If you have any more feedback please give it to me here or on the Draft page, its talk page or my user talk page. Thank you for helping me! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:20, 18 April 2026 (UTC) ::Today I woke up with not only thinking about supplying questions along with the "idea" but also answers. ie. Is it possible to "test" this idea? Is it possible to create one or multiple hypotheses based on this "idea"?(etc.) I've thought about this before in this "idea" but since I'm beginning to add to Wikiversity what was previously 'locked in my mind' it's also easier for me to see what I've done so far. Thank you for this comment! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 09:11, 23 April 2026 (UTC) :May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish. Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet. Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible. Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, "Questionable Language (profanity)", which I don't understand…). My few cents. -- [[User:Eric.LEWIN|Eric.LEWIN]] ([[User talk:Eric.LEWIN|discussion]] • [[Special:Contributions/Eric.LEWIN|contributions]]) 10:06, 17 April 2026 (UTC) ::Sorry about the false positive on the profanity filter - I've fixed it. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:26, 17 April 2026 (UTC) :::"May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish." ::Thank you Eric for this comment. Trust in time is how I interpret it. I should not feel like I need to be in a hurry. I'll try to give this time. Thank you! :::"Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet." ::A central place for developing or making "project notes" regarding the Neurodiversity idea on my userspace, I might need that, like a diary or "project notes" of the Neurodiversity idea similar to my course notes regarding my experience with Coursera. ::Any actions I take are going to be related to my Userspace from now on but I'll also update the draft when necessary. Now in the beginning I might be working daily to once every 3 days on both the draft and the daily notes I plan to make. :::"Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible." ::Thank you for the advice. I was brainstorming yesterday about it. I concluded that since I've not yet developed a methodology that adheres to "Do no harm" and this is my first time working my "idea" into a way that is compatible with how projects develop on English Wikiversity this is new to me. My methodology isn't developed and therefore trying to get attention to my project through a name can wait. Yesterday I figured out a silly title that has nothing to do with the project: "Planetary Awareness Potato Cabbage Rolls" or something like that. Google output read that no such thing exists so I wanted it mainly to be unique. I don't want to raise attention that I'm unsure whether I'll actually be capable of developing a methodology for but project notes is my best bet so far in tracking my progress. Every day I think about this "idea" but I need to improve the important parts. :::"Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, "Questionable Language (profanity)", which I don't understand…). My few cents." ::You added great points and I felt that I was helped by you! I encourage you to post again and I can understand that interacting with any kind of automated filter can be discouraging and can be for me too! Thank you for giving me feedback! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 16:01, 18 April 2026 (UTC) == Add some user rights to the curator user group? == By default, only custodians have the ability to mark new pages as patrolled (<code>patrol</code>) and have their own page creations automatically marked as patrolled (<code>autopatrol</code>). I am proposing both of the following: * Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages. * New pages made by curators will be automatically marked as patrolled by the MediaWiki software. Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC) :Agree, <s>also can we also allow curators to undelete pages since they already have the rights to delete them?</s> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC) ::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC) ::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC) ::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC) :{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 14:35, 18 April 2026 (UTC) :'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC) : I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC) == Wikiversity:Curators to become a policy == I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC) :{{support}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 18 April 2026 (UTC) :{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC) : {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC) == Inactive curators == Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more: * {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022) * {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022) [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC) :Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC) :: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC) == Is anyone interested in Neurodiversity? == Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be "something more" than it already is? Does anyone here consider Neurodiversity one of the "harder topics" to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like "out of a vacuum" I can tell you a bit about my background: Many years ago I got a psychiatric diagnosis "Asperger's". After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of "labeling others". I was unsure whether this was "real science" and sort of "challenged myself" to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing "patterns". Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how "their brains work". Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the "chaos theory". He told me about "chaos theory" while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is "did he understand about me what I think I understood about him?"? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry. I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's "neurotypes" interested me for a while until I realized I had "misunderstood everything" about them and how they are used in the Neurodiversity Movement or "Neurodiversity community" if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page. So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a "comprehensive way" or show that it matters to me nothing is going to ever happen. So I started talking about my "idea" more. Nobody could understand the "idea" because I had not developed my skills regarding where to start...although the process had already started "automatically" and that's why I often think of "well my brain sort of activated me". I don't feel like I did have a plan and this idea happened. It happened "by itself". My brain reacted to what I was seeing in a video or stream. I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like "don't analyze me!", "don't compare me to anyone!". On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a "science" way if done the "right way" which probably means "Do no harm". I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a "reasonable" use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what "group configurations" help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to "give back" to the state and "the world". May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC) == Request for comment (global AI policy) == <bdi lang="en" dir="ltr" class="mw-content-ltr"> A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC) </bdi> <!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30424282 --> qah5autxnwmllyk7y3qjdjg7lml31sx 2806776 2806681 2026-04-27T10:12:55Z ThinkingScience 3061446 /* Why does it feel like Wikiversity is no longer really active anymore? */ Reply 2806776 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Requested update to [[Wikiversity:Interface administrators]] == Currently, [[Wikiversity:Interface administrators]] is a policy that includes a caveat that interface admins are not required long-term and that user right can only be added for a period of up to two weeks. I am proposing that we remove this qualification and allow for indefinite interface admin status. I think this is useful because there are reasons for tweaking the site CSS or JavaScript (e.g. to comply with dark mode), add gadgets (e.g. importing Cat-a-Lot, which I would like to do), or otherwise modifying the site that could plausibly come up on an irregular basis and requiring the overhead of a bureaucrat to add the user rights is inefficient. In particular, I am also going to request this right if the community accepts indefinite interface admins. Thoughts? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:23, 17 August 2025 (UTC) :And who will then monitor them to make sure they don't damage the project in any way, or abuse the rights acquired in this way? For large projects, this might not be a problem, but for smaller projects like the English Wikiversity, I'm not sure if there are enough users who would say, something is happening here that shouldn't be happening. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:28, 20 August 2025 (UTC) ::Anyone would be who. This argument applies to any person with any advanced rights here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:46, 20 August 2025 (UTC) :I think it is reasonable to allow for longer periods of access than 2 weeks to interface admin and support adjusting the policy to allow for this flexibility. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:57, 2 December 2025 (UTC) ::+1 —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:38, 25 January 2026 (UTC) :@[[User:Koavf|Koavf]] I agree that the two-week requirement could be revised, but wouldn’t people just request access for a specific purpose anyway? Instead of granting indefinite access, they should request the specific time frame they need the rights for—until the planned fixes are completed—and then request an extension if more time is required. We could remove the two-week criterion while still keeping the access explicitly temporary. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:48, 25 January 2026 (UTC) ::I just don't see why this wiki needs to be different than all of the others. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:18, 25 January 2026 (UTC) :::There isn’t really much of a need for a permanent one at this point in time [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 25 January 2026 (UTC) :I quite agree with this proposal, so long as they perform the suggested changes as mentioned here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 04:06, 26 January 2026 (UTC) :: Just to clarify, I support '''indefinite interface admin status'''. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:34, 13 April 2026 (UTC) :I think there is decent consensus for lengthening this, but not necessarily for indefinite permissions, so does anyone object to me revising it to the standard being 120 days instead of two weeks? I'll check back on this thread in three weeks and if there's no objection, I'll make the change. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:47, 13 April 2026 (UTC) ::Sure [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:27, 13 April 2026 (UTC) ::Thanks for proposing this, Justin. I agree with the proposal to lengthen the interface admin period from 2 weeks but not indefinitely. Can I check the source(s) for the standard being 120 days (I'm guessing policies on other projects or maybe global policy?)? In any case, I think it is reasonable for us to adopt a similar period. However, note on the current policy discussion page notes from @[[User:Dave Braunschweig|Dave Braunschweig]] arguing for shorter periods to lower risk, that's why it is 2 weeks. But if there are projects that need longer access, that should also be accommodated. Maybe we could adjust the policy to specify that ''interface admin rights can be given for 14 to 120 days depending on how long is required and what is supported by the community''. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:29, 24 April 2026 (UTC) :::There was there was no source for 120: it was just more than 14 and less than infinity. The "14 to 120" also seems reasonable. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 14:33, 24 April 2026 (UTC) ::: On some small/medium-sized wikis, such as English Wikibooks and English Wikiquote for example, indefinite interface administrator access for administrators is allowed, but they tend not to make changes to the CSS and JS page changes unless it's truly necessary. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:34, 24 April 2026 (UTC) :::It's a good idea to make the length of this right on request or allow to be prolonged. However, IA should test large changes somewhere else, for example on the en.wv mirror, and only after testing it on the mirror, adapt it to the live version. That means I can't imagine a time-consuming operation right now. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:04, 24 April 2026 (UTC) ::::Sorry, what mirror is this? Are you talking about beta.wv? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:32, 24 April 2026 (UTC) :::::Not beta.wv. Basically somewhere else then on a live wiki. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:59, 24 April 2026 (UTC) :::::: Wouldn't testing on a user's own common.css page work anyway? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:36, 24 April 2026 (UTC) == [[Wikiversity:Curators|Curators and curators policy]] == How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC) :It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|<span style="font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;">RailwayEnthusiast2025</span>]] ^{[[User talk:RailwayEnthusiast2025|<span style="color:#59a53f">''talk with me!''</span>]]} 18:33, 27 October 2025 (UTC) :I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:08, 3 December 2025 (UTC) :What? I thought you were getting it approved, Juandev... :) [[User:I&#39;m Mr. Chris|I&#39;m Mr. Chris]] ([[User talk:I&#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC) ::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC) :::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC) ::::Agree -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:00, 27 March 2026 (UTC) Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:40, 24 April 2026 (UTC) == [[Template:AI-generated]] == After going through the plethora of ChatGPT-generated pages made by [[User:Lbeaumont|Lbeaumont]] (with many more pages to go), I'd like community input on this proposal to [[Wikiversity:Artificial intelligence]] that I think would be benefical for the community: *Resources generated by AI '''must''' be indicated as so through the project box, [[Template:AI-generated]], on either the page or the main resource (if the page is a part of a project). I do not believe including a small note/reference that a page is AI-generated is sufficient, and I take my thinking from [[WV:Original research|Wikiversity's OR policy]] for OR work: ''Within Wikiversity, all original research should be clearly identified as such''. I believe resources created from AI should also be clearly indicated as such, especially since we are working on whether or not AI-generated resources should be allowed on the website (discussion is [[Wikiversity talk:Artificial intelligence|here]], for reference). This makes it easier for organizational purposes, and in the event ''if'' we ban AI-generated work. I've left a message on Lee's talk page over a week ago and did not get a response or acknowledgement, so I'd like for the community's input for this inclusion to the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 26 January 2026 (UTC) :I believe that existing Wikiversity policies are sufficient. Authors are responsible for the accuracy and usefulness of the content that is published. This policy covers AI-generated content that is: 1) carefully reviewed by the author publishing it, and 2) the source is noted.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:38, 27 January 2026 (UTC) ::A small reference for pages that are substantially filled with Chat-GPT entries, like [[Real Good Religion]], [[Attributing Blame]], [[Fostering Curiosity]], are not sufficient IMO and a project box would be the best indicator that a page is AI-generated (especially when there is a mixture of human created content AND AI-generated content, as present in a lot of your pages). This is useful, especially considering the notable issues with AI (including hallucinations and fabrication of details), so viewers and support staff are aware. These small notes left on the pages are not as easily viewable as a project box or banner would be. I really don't see the issue with a clear-label guideline. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:34, 27 January 2026 (UTC) ::{{ping|Lbeaumont}} I noticed your reversions [https://en.wikiversity.org/w/index.php?title=Exploring_Existential_Concerns&diff=prev&oldid=2788278 here] & [https://en.wikiversity.org/w/index.php?title=Subjective_Awareness&diff=prev&oldid=2788257 here]. I'd prefer to have a clean conversation regarding this proposition. Please voice your concerns here. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 28 January 2026 (UTC) :::Regarding Subjective Awareness, I distinctly recall the effort I went to to write that the old-fashioned way. It is true that ChatGPT assisted me in augmenting the list of words suggested as candidate subjective states. This is a small section of the course, is clearly marked, and makes no factual claim. Marking the entire course as AI-generated is misleading. I would have made these comments when I reverted your edit; however, the revert button does not provide that opportunity. :::Regarding the Exploring Existential Concerns course, please note this was adapted from my EmotionalCompetency.com website, which predates the availability of LLMs. The course does include two links, clearly labeled as ChatGPT-generated. Again, marking the entire course as AI-generated is misleading. :::On a broader issue, I don't consider your opinions to have established a carefully debated and adopted Wikiversity policy. You went ahead and modified many of my courses over my clearly stated objections. Please let this issue play out more completely before editing my courses further. Thanks.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:11, 29 January 2026 (UTC) ::::Understood, and I respect your position. I apologize if my edits were seen as overarching. We could change the project box to "a portion of this resource was generated by AI", or something along those lines. Feel free to revert my changes where you see fit, and I encourage more users to provide their input. EDIT: I've made changes to the template to indicate that a portion of the content has been generated from an LLM. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:50, 29 January 2026 (UTC) :::::Thanks for this reply. The new banner is unduly large and alarming. There is no need for alarm here. The use of AI is not harmful per se. Like any technology, it can be used to help or to harm. I take care to craft prompts carefully, point the LMM to reliable source materials, and to carefully read and verify the generated text before I publish it. This is all in keeping with long-established Wikiversity policy. We don't want to use a  [[w:One-drop_rule|one-drop rule]] here or cause a [[w:Satanic_panic|satanic panic]]. We can learn our lessons from history here. I don't see any pedagogical reason for establishing a classification of "AI generated", but if there is a consensus that it is needed, perhaps it can be handled as just another category that learning resources can be assigned to. I would rather focus on identifying any errors in factual claims than on casting pejorative bias toward AI-generated content. An essay on the best practices for using LMM on Wikiveristy would be welcome.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:58, 30 January 2026 (UTC) ::::::The new banner mimics the banner that is available on the English Wikibooks (see [[b:Template:AI-generated]] & [[b:Template:Uses AI]]), so my revisions aren't unique in this aspect. At this point, I'd welcome other peoples' inputs. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:40, 30 January 2026 (UTC) == How do I start making pages? == Is there a notability guideline for Wikiversity? What is the sourcing policy for information? What is the Manual of Style? What kind of educational content qualifies for Wikiversity? All the introduction pages are a bit unclear. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 02:25, 28 January 2026 (UTC) :{{ping|VidanaliK}} Welcome to Wikiversity! I've left you a welcome message on your talk page. That should help you out. Make sure to especially look at [[Wikiversity:Introduction]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:11, 28 January 2026 (UTC) ::It says that I can't post more pages because I have apparently exceeded the new page limit. How long does it take before that new page limit expires? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 16:57, 28 January 2026 (UTC) :::This is a restriction for new users so that Wikiversity is not hit with massive spam. As for when this limit will expire, it should be a few days or after a certain number of edits. It's easy to overcome, though I do not have the exact numbers atm. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:08, 29 January 2026 (UTC) ::::OK, I think I got past the limit. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 17:21, 29 January 2026 (UTC) ==Why does it feel like Wikiversity is no longer really active anymore?== I've been looking at recent changes, and both today and yesterday there haven't been many changes that I haven't made; it feels like walking through a ghost town, is this just me or is Wikiversity not really active anymore? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 03:54, 30 January 2026 (UTC) :There is fewer people editing these days compared to the past. Many newcomers tend to edit in Wikipedia instead. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 06:39, 30 January 2026 (UTC) :It’s a little slow, but I’m happy to know that Wikiversity is a place that I think should provide value even if the activity of editors fluctuates. If it’s any consolation your edits may be encouraging for some anonymous newcomer to start edits on their own! I think it’s hard to build community when there is such a wide variety of interests and a smaller starting userbase. Also sometimes the getting into a particular topic that already exists can be intimidating because some relics (large portals, school, categories, etc.) have intricate, unique and generally messy levels of organization. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:16, 9 March 2026 (UTC) :I'd say it comes down to working hard for Wikiversity, basically if somebody or a group of people will start presenting good ideas and they turn out to be provably stable. :I even asked Google's "AI Mode", what is Wikiversity famous for? Unfortunately it could not answer that. :Simply, we have not made Wikiversity famous by presenting really provable stable ideas yet. My hope is that this time might come. Perhaps even this year 2026! :Hope dies last. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:12, 27 April 2026 (UTC) == Inactivity policy for Curators == I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC) :Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC) ::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC) :::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC) ::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC) ::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]: ::::::The maximum time period of inactivity <u>without community review</u> for curators is two years (consistent with the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights. ::::: -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:51, 27 March 2026 (UTC) :::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC) :::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC) :I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:59, 24 April 2026 (UTC) :: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC) == [[Wikiversity:Artificial intelligence]] to become an official policy == {{Archive top|After running for a week, there is consensus, alongside comments, for [[Wikiversity:Artificial intelligence]] to be implemented as an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:27, 17 April 2026 (UTC)}} With the introduction of AI-material, and some material just plain disruptive, its imperative that Wikiversity catches up with its sister projects and implements an official AI policy that we can work with. The recent issue of [[User:Lbeaumont|Lbeaumont]]'s 50+ articles that contain significantly large AI-generated material has made me came to the Colloquium. This user has also been removing the [[Template:AI-generated]] template from their pages, calling it "misleading", "alarmist", and "pejorative" - which is all just simply nonsensical rationales. Not to even mention this user's contributions to the English Wikipedia have been [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Inner_Development_Goals contested] and [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Multipolar_trap removed] a couple of times (for being low-quality and clearly LLM-generated), highlighting the need for an actual policy to be implemented here on Wikiversity. I would like to ping {{ping|Juandev}} and {{ping|Jtneill}} for their thoughts as well, since I'd like this to be implemented as soon as possible. Wikiversity has a significant issue with implementing anti-disruptive measures, hence why we have received numerous complaints as a community about our quality. I originally was reverting the removal of the templates, but realized that this is still a proposed policy, which it shouldn't be anymore. It should be a recognized Wikiversity policy. 14:54, 10 March 2026 (UTC) —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:54, 10 March 2026 (UTC) :@[[User:Atcovi|Atcovi]] '''I agree''' that the draft, should become official policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:00, 10 March 2026 (UTC) :I provided a detailed response at: [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI]] :I will appreaciate it if you consder that carefully. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 22:49, 10 March 2026 (UTC) :Agree it should become official Wikiversity policy on the condition <u>that point point 5 is about [significant/substantial] LLM-generated text specifically</u>. Not a good idea to overuse it, it should be added when there is substantial AI-generated text on the page, not for other cases. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:37, 11 March 2026 (UTC) :What policy is being debated? Is it the text on this page, which is pointed to by the general banner, or the text at:   [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence,]]   which is pointed to by the specific banner? Let's begin with coherence on the text being debated. Thanks! [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:49, 17 March 2026 (UTC) ::@[[User:Lbeaumont|Lbeaumont]] This is a call for approval of the new Wikiversity policy. You expressed your opinion [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI|on the talk page of the proposal]], I replied to you and await your response.When creating policies, it is necessary to propose specific solutions. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:12, 17 March 2026 (UTC) :::Toward a Justified and Parsimonious AI Policy :::As we collaborate to develop a consensus policy on the use of Large Language Models, it is wise to begin by considering the needs of the various stakeholders to the policy. :::The stakeholders are: :::1)     The users, :::2)     The source providers, and :::3)     The editors :::There may also be others with a minor stake in this policy, including the population at large. :::The many needs of the users are currently addressed by long-standing [[Wikiversity:Policies|Wikiversity policies]], so we can focus on what, if any, additional needs arise as LLMs are deployed. :::As always, users need assurance that propositional statements are accurate. This is covered by the existing policy on [[Wikiversity:Verifiability|verifiably]]. In addition, it is expected by both the users and those that provide materials used as sources for the text are [[Wikiversity:Cite sources|accurately attributed]]. This is also covered by [[Wikiversity:Cite sources|existing policies]]. :::To respect the time and effort of editors, a parsimonious policy will unburden editors from costly requirements that exceed benefits to the users. :::Finally, it is important to recognize that because attention is our most valuable seizing attention unnecessarily is a form of theft. :::The following proposed policy statement results from these considerations: :::Recommended Policy statement: :::·       Editors [[Wikiversity:Verifiability|verify the accuracy]] of propositional statements, regardless of the source. :::·       Editors [[Wikiversity:Cite sources|attribute the source]] of propositional statements. In the case of LLM, cite the LLM model and the prompt used. :::·       Use of various available templates to mark the use of LLM are optional. Templates that are flexible in noting the type and extend of LLM usage are preferred. Templates that avoid unduly distracting or alarming the user are preferred. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:56, 19 March 2026 (UTC) ::::Do we discuss here or there? I have replied you there as your proposal is about that policy so it is tradition to discuss it at the affected talk page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:59, 19 March 2026 (UTC) : {{support}} Thanks for the proposed policy development and discussion; also note proposed policy talk page discussion: [[Wikiversity talk:Artificial intelligence]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:05, 24 March 2026 (UTC) ::I think the Wikiversity AI policy shall be official. – [[User:RestoreAccess111|RestoreAccess111]] <sup style="font-family:Arimo, Arial;">[[User talk:RestoreAccess111|Talk!]]</sup> <sup style="font-family:Times New Roman, Tinos;">[[Special:Contributions/RestoreAccess111|Watch!]]</sup> 06:11, 13 April 2026 (UTC) {{archive bottom}} == New titles for user right nominations == <div class="cd-moveMark">''Moved from [[Wikiversity talk:Candidates for Custodianship#New titles for user right nominations]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 17 April 2026 (UTC)''</div> I would like to propose the following retitles should a user be nominated for any of the following user rights: * Curator: Candidates for Curatorship * Bureaucrat: Candidates for Bureaucratship The reason is that many curator (and probably bureaucrat) requests have run solely under {{tq|Candidates for Custodianship}}, but that title might sound misleading (especially in regards to the permission a user is requesting). CheckUser and Oversight (suppressor) are not included above since no user was nominated for these sensitive permissions, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:30, 19 March 2026 (UTC) :And it's not that when someone at the beginning misplaced the request, no one thought to move it and the others copied it. Even today, it would be possible to simply take it all and move it. Otherwise, for me, the more fundamental problem is that there is [[Wikiversity:Curators|no approved policy for curators]] than where the requests are based. Curators then operate in a certain vacuum and if one of them "breaks out of the chain", the average user doesn't have many transparent tools to deal with it, because there is no policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:02, 19 March 2026 (UTC) ::I am not talking about the curator page (policy proposal). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:08, 21 March 2026 (UTC) : @[[User:Juandev|Juandev]] I'll see if I can do an overhaul of [[Wikiversity:Candidates for Custodianship]], just like I recently did with the Requests for adminship page on English Wikiquote. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:17, 18 April 2026 (UTC) == Technical Request: Courtesy link.. == [[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC) : I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC) :: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :) [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC) :I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC) : I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC) == WikiEducator has closed == Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/]. It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating. They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki. The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license). The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 1 April 2026 (UTC) :I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:46, 1 April 2026 (UTC) == Wikinews is ending == Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]). And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC) :Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well. :In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up "math properties" for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). :I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC) :For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest "lagging behind" or "failure" project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:46, 2 April 2026 (UTC) :[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:03, 11 April 2026 (UTC) == Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) == Hello everyone, This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as <code>Module:Itemgroup/list</code>). '''The Change:'''<br /> Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''. '''What You Need To Do:'''<br /> To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search. '''Deadline:'''<br /> We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles. Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:11, 3 April 2026 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=29941252 --> :I didnt find such properties, so we are probably fine. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:00, 12 April 2026 (UTC) :: +1 (agreed). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:19, 12 April 2026 (UTC) == Enable the abuse filter block action? == In light of [[Special:AbuseLog/80178]] (coupon spam), I would like to propose enabling the block action for the abuse filter. Only custodians will be able to enable and disable that action on an abuse filter, and it is useful to block ongoing vandalism. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:12, 13 April 2026 (UTC) :Seems like a good idea, almost all of the users which create such pages are spambots so this shouldn’t be a problem. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:41, 13 April 2026 (UTC) :Can you explain some more (I am new to abuse filters)? It looks like the attempted edit was prevented? Which abuse filter? :Note on your suggestion, have also reactivated Antispam Filter 12 - see [[WV:RCA]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:45, 15 April 2026 (UTC) :: I am proposing that we activate the abuse filter block action, which if a user triggers an abuse filter, it would actually block the user in question - the same mechanism that a custodian would use to block users. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:11, 15 April 2026 (UTC) :::OK, thankyou, that makes sense. And, reviewing the abuse filter 12 log, it would be helpful because it would prevent the need for manual blocking. But I don't see a setting for autoblocking? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:14, 15 April 2026 (UTC) :::: I think it probably adds an autoblock. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:43, 16 April 2026 (UTC) : [[User:Jtneill|Jtneill]] and [[User:PieWriter|PieWriter]], given that a little bit more than a week has passed and there is minimal consensus to activate the abuse filter block action, I filed [[phab:T424053]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:05, 21 April 2026 (UTC) ::Thank-you for doing this. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:03, 24 April 2026 (UTC) == Advice needed: A Neurodiversity-inspired Idea/observation == If I want the greatest participation of others to "provide constructive criticism to my idea" or to "shoot down my idea" or "idea". What I've called it so far is "The Neurodiversity-inspired Idea". At other times I used more sensationalist wording but here on Wikiversity I don't dare do that. I actually woke up with thinking about putting this into my userspace draft: "Personal Observations Made By Meeting Autistic and Non-Autistic Adults". My ultimate goal is to stop blathering about my "idea" to friend and family without feeling my "methodology" is going into any progressive direction whatsoever. My latest encounter was somewhat constructive though. A friend of a friend who worked with people presenting ideas in attempting to getting grants. I don't want a grant. I just want to figure out how I can express my "idea" in a way so that I can more clearly figure out what flaws it got. At the same time I tend to overthink. If anyone thinks etherpad might be a good place and considering Wikimedia already got an etherpad at https://etherpad.wikimedia.org/ if anyone feels like they know me better in the future feel free to suggest a "session" on etherpad. '''If I don't receive a reply to this in 1 week's time I will begin to explore this "idea" into my userspace''' unless you replied and refrained me from doing so, of course. Then maybe after "developing it there" I might reference it to you another future time here in the Colloquium, with my "idea" still in my userspace draft. This "idea" is sort of a burden, I'm happy I've made the choice to get rid of it and hopefully move on with my life, unless there is something to this "idea". My failure is probably evident: I feel I haven't told you anything. Same happened to when I talked to friends and family. In danger of overthinking it further I'll publish this right now. I need to "keep it together" [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:36, 16 April 2026 (UTC) :Good on you putting it out there ... and hitting publish :). I'd say go for it (no need to wait), give birth to your idea and share about it here and elsewhere. Let it take shape and see where it might go. In many ways, this is exactly what an open collaborative learning community should be doing. Others might not know well how to respond, so perhaps consider creating some questions to accompany the idea. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:21, 16 April 2026 (UTC) ::Thank you for encouraging me in developing the idea. ::I have created a "questions" section in the draft which is visible in the table of contents now. My brain was "frozen" today metaphorically speaking in that I felt I had like a "writer's block" so the draft has more "AI/LLM" content than before. I used the LLM for generating questions. The answers are so far human-only. ::I've also created a subsection where I could add the prompts that made the LLM generate the questions. That could help people make better prompts perhaps. I've described what it is about inside of it and there are some chaotically written notes. ::[[Draft:The_Neurodiversity-inspired_Idea#Questions_that_might_encourage_the_development_of_this_idea_and_its_methodology]] ::My draft is missing stuff. Any questions that you contribute to my draft will probably help me and if I don't understand the questions I'll probably notify you and also at the same time "feed them" to an LLM and ask in my input like "explain in simple words what this question means, what is it searching for?" etc. while I wait for an answer. If you have any more feedback please give it to me here or on the Draft page, its talk page or my user talk page. Thank you for helping me! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:20, 18 April 2026 (UTC) ::Today I woke up with not only thinking about supplying questions along with the "idea" but also answers. ie. Is it possible to "test" this idea? Is it possible to create one or multiple hypotheses based on this "idea"?(etc.) I've thought about this before in this "idea" but since I'm beginning to add to Wikiversity what was previously 'locked in my mind' it's also easier for me to see what I've done so far. Thank you for this comment! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 09:11, 23 April 2026 (UTC) :May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish. Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet. Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible. Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, "Questionable Language (profanity)", which I don't understand…). My few cents. -- [[User:Eric.LEWIN|Eric.LEWIN]] ([[User talk:Eric.LEWIN|discussion]] • [[Special:Contributions/Eric.LEWIN|contributions]]) 10:06, 17 April 2026 (UTC) ::Sorry about the false positive on the profanity filter - I've fixed it. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:26, 17 April 2026 (UTC) :::"May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish." ::Thank you Eric for this comment. Trust in time is how I interpret it. I should not feel like I need to be in a hurry. I'll try to give this time. Thank you! :::"Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet." ::A central place for developing or making "project notes" regarding the Neurodiversity idea on my userspace, I might need that, like a diary or "project notes" of the Neurodiversity idea similar to my course notes regarding my experience with Coursera. ::Any actions I take are going to be related to my Userspace from now on but I'll also update the draft when necessary. Now in the beginning I might be working daily to once every 3 days on both the draft and the daily notes I plan to make. :::"Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible." ::Thank you for the advice. I was brainstorming yesterday about it. I concluded that since I've not yet developed a methodology that adheres to "Do no harm" and this is my first time working my "idea" into a way that is compatible with how projects develop on English Wikiversity this is new to me. My methodology isn't developed and therefore trying to get attention to my project through a name can wait. Yesterday I figured out a silly title that has nothing to do with the project: "Planetary Awareness Potato Cabbage Rolls" or something like that. Google output read that no such thing exists so I wanted it mainly to be unique. I don't want to raise attention that I'm unsure whether I'll actually be capable of developing a methodology for but project notes is my best bet so far in tracking my progress. Every day I think about this "idea" but I need to improve the important parts. :::"Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, "Questionable Language (profanity)", which I don't understand…). My few cents." ::You added great points and I felt that I was helped by you! I encourage you to post again and I can understand that interacting with any kind of automated filter can be discouraging and can be for me too! Thank you for giving me feedback! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 16:01, 18 April 2026 (UTC) == Add some user rights to the curator user group? == By default, only custodians have the ability to mark new pages as patrolled (<code>patrol</code>) and have their own page creations automatically marked as patrolled (<code>autopatrol</code>). I am proposing both of the following: * Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages. * New pages made by curators will be automatically marked as patrolled by the MediaWiki software. Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC) :Agree, <s>also can we also allow curators to undelete pages since they already have the rights to delete them?</s> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC) ::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC) ::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC) ::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC) :{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 14:35, 18 April 2026 (UTC) :'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC) : I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC) == Wikiversity:Curators to become a policy == I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC) :{{support}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 18 April 2026 (UTC) :{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC) : {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC) == Inactive curators == Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more: * {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022) * {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022) [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC) :Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC) :: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC) == Is anyone interested in Neurodiversity? == Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be "something more" than it already is? Does anyone here consider Neurodiversity one of the "harder topics" to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like "out of a vacuum" I can tell you a bit about my background: Many years ago I got a psychiatric diagnosis "Asperger's". After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of "labeling others". I was unsure whether this was "real science" and sort of "challenged myself" to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing "patterns". Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how "their brains work". Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the "chaos theory". He told me about "chaos theory" while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is "did he understand about me what I think I understood about him?"? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry. I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's "neurotypes" interested me for a while until I realized I had "misunderstood everything" about them and how they are used in the Neurodiversity Movement or "Neurodiversity community" if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page. So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a "comprehensive way" or show that it matters to me nothing is going to ever happen. So I started talking about my "idea" more. Nobody could understand the "idea" because I had not developed my skills regarding where to start...although the process had already started "automatically" and that's why I often think of "well my brain sort of activated me". I don't feel like I did have a plan and this idea happened. It happened "by itself". My brain reacted to what I was seeing in a video or stream. I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like "don't analyze me!", "don't compare me to anyone!". On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a "science" way if done the "right way" which probably means "Do no harm". I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a "reasonable" use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what "group configurations" help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to "give back" to the state and "the world". May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC) == Request for comment (global AI policy) == <bdi lang="en" dir="ltr" class="mw-content-ltr"> A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC) </bdi> <!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30424282 --> prvvigo2lemr6sz1px9u2nje4k3zcmg 3 4866 2806782 2804690 2026-04-27T11:45:48Z ThinkingScience 3061446 /* Different way to display talk pages for easier reading? */ new section 2806782 wikitext text/x-wiki {| style="border-spacing:8px;margin:0px -8px" width="100%" |class="MainPageBG" style="width: 55%; border:1px solid #084080; background-color:#F5FFFA; vertical-align:top;color:#000000;font-size: 85%"| {| width="100%" cellpadding="2" cellspacing="5" style="vertical-align:top; background-color:#F5FFFA" ! <div style="margin: 0; background-color:#CEF2E0; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #084080; text-align:left; color:#082840; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;"> '''Hello Koavf! [[Wikiversity:Welcome, newcomers|Welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you decide that you need help, check out [[Wikiversity:Help desk]], ask the [[Wikiversity:Support staff|support staff]], or ask me on my talk page. Please remember to [[Wikiversity:Sign your posts on talk pages|sign your name]] on talk pages using four tildes (~&#126;~~); this will automatically produce your name and the date. Below are some recommended guidelines to facilitate your involvement. Happy Editing! -- [[User:Trevor MacInnis|Trevor MacInnis]] 22:28, 4 September 2006 (UTC)</div> |} {| style="border-spacing:8px;margin:0px -8px" width="100%" |class="MainPageBG" style="width: 55%; border:1px solid #FFFFFF; background-color:#F5FFFA; vertical-align:top"| {| width="100%" cellpadding="2" cellspacing="5" style="vertical-align:top; background-color:#F5FFFA" ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #CEF2E0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting Started</div> |- |style="color:#000"| * [[Wikiversity:Guided tour|Take a guided tour]] * [[Help:Editing|How to edit a page]] * [[Wikiversity:Be bold|Be bold in editing]] * [[Portal:Learning Projects|Learning Projects]] * [[Wikiversity:What Wikiversity is not|What Wikiversity is not]] |- ! <div style="margin: 0; background:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting your info out there</div> |- | style="color:#000"| * [[Wikiversity:Cite sources|Cite your sources]] * [[Wikiversity:Disclosures|Neutral Point of View]] * [[Wikiversity:Verifiability|Verifiability]] |- ! <div style="margin: 0; background:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting more Wikiversity rules</div> |- | style="color:#000"| * [[Wikiversity:Policies|Policy Library]] |- |} |class="MainPageBG" style="width: 55%; border:1px solid #FFFFFF; background-color:#F5FFFA; vertical-align:top"| {| width="100%" cellpadding="2" cellspacing="5" style="vertical-align:top; background-color:#F5FFFA" ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #CEF2E0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting Help</div> |- |style="color:#000"| * [[Wikiversity:Research|Research guidelines]] * [[Wikiversity:Help desk|Help Desk]] |- ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting along</div> |- |style="color:#000"| * [[Wikiversity:Civility|Civility]] * [[Wikiversity:Sign your posts on talk pages|Sign your posts]] * [[Wikiversity:Scholarly ethics|Scholarly ethics]] |- ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting technical</div> |- |style="color:#000"| [[Image:Wikimedia Foundation RGB logo with text.svg|60px|right]] * [[Wikiversity:Colloquium|Colloquium]] |- |} |} |} == wikitravel == Hi. You removed links to Wikitravel. Why? --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 12:44, 24 October 2013 (UTC) :'''Wikitravel links''' Per discussion at [[w:Template:Wikitravel|en.wp]] as well as [[m:Interwiki map|Meta]] to remove links at those projects. If you want to keep links and references here at en.v, I guess that's fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:28, 24 October 2013 (UTC) == Thanks. == I see you got it before I explained. Wikiversity is disconcerting to those familiar with the encyclopedia projects, and the other content-oriented projects. While we do have a content mission, we ''also'' have a "learning by doing" mission, which is about ''people.'' Our product is not just content, it is education, and there is no education without users who are educated, and sophisticated education is always about process and people skills and the rest. I would argue that the encyclopedia projects also need to be welcoming, if the full mission is to be fulfilled, but ... they developed with a very narrow focus and absent the realization that an environment that was easily seen as hostile would damage the mission. The 20th century saw the development of systems and skills and process for maximizing consensus, and the only reliable measure of neutrality is level of consensus. (I.e., if everyone involved agrees, 100% consensus, while what they agree upon only might possibly turn out, in the end, to be defective or invalid, there is no better measure!). So to the extent that there is exclusion, to that extent, the assessment of neutrality can be warped. Obviously, compromises are necessary, but "compromise" requires tolerating a level of damage, and that is easily forgotten. When the importance of consensus being as broad as possible is realized, a community will find ways to keep conversation open, on some level, in some place, otherwise the community becomes locked into what I call the "tyranny of the past." There is a children's song that was part of a therapeutic response to Reactive Attachment Disorder: :'''There is always something you can do, do, do''' :'''When you're getting in a stew, stew, stew.''' Mostly, it involves simmering down, dropping upset and reactive response, and, when calm, communicating. While this kind of work has been done on Wikipedia, often in user space -- it's what I did, successfully mediating disputes, such that users at each other's throats became cooperative ''with each other'' -- this was mumbo-jumbo nonsense to too many on Wikipedia. For example, see [https://en.wikipedia.org/wiki/Wikipedia:Miscellany_for_deletion/Abd_user_pages], which included many pages of historical function, including evidence presented to ArbCom. I found it very strange that ArbCom did not care that evidence used in a case was being deleted, but ArbCom consists of too many elevated beyond their competence by popularity (as well as many other highly-experienced and thoughtful user; but the system tends to burn them out and they become less attentive.) [[w:User:Abd/Dispute over thermoeconomics]] was particularly educational. In that mediation, a professor was revert warring with Randy from Boise, so to speak, and one or both were about to be blocked. It took very little to develop cooperation, mostly just sitting them down together with some support. Hmmm... I'm thinking of asking that these pages be transwikied to Wikiversity, precisely for historical study. Looking for the link to that, I came across [https://en.wikipedia.org/wiki/Wikipedia:Miscellany_for_deletion/User:UBX/Esperanza_returns this]. It shows a quick and major clue to what happened on en.wiki. Two three-letter users with a conflict. One was an administrator taken to ArbCom by the other, and the administrator was trout-slapped by ArbCom and then, it is obvious, revenge was exacted, by the admin and his friends. This was long-continued and, while not unnoticed, never sanctioned. Admins can be hostile, this one was more than hostile, he was highly insulting at times, using obscene language, and using tools while involved, was reprimanded, made small adjustments to his behavior, but continued pretty much unimpeded. And, as you know, this is not uncommon. He is even a likeable Guy. I consider this all the responsibility ''of the community.'' Blaming people for what comes naturally for them is not productive. Such people generally will modify behavior in a functional community. Notice the irony. The userbox was "Esperanza returns," referring to the project designed to foster civility and welcome and cooperation. Esperanza, of course, means Hope. So the nominator was saying, "Hope will never return." Esperanza was crushed when it temporarily was inactive. Instead of improving the governance, which was easily possible, it was crushed with ''vehemence,'' see the [[w:Wikipedia:Miscellany_for_deletion/Wikipedia:Esperanza|MfD]]. Why? To any serious student of human organizational structure, it's obvious. Wikiversity is the slim thread of hope, and if it is not protected and defended, hope will break. Thanks again. --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 15:17, 7 August 2015 (UTC) == Curator == Hi! I've noticed and appreciated your recent efforts on behalf of Wikiversity. Do you have any interest in becoming a [[Wikiversity:Curators|Wikiversity curator]]? It would give you additional tools to make some clean-up easier. I'd be happy to nominate/support you if you are interested. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:11, 19 October 2016 (UTC) :{{Ping|Dave Braunschweig}} I'd be delited and honored. I started editing here as soon as it was founded and I've always wanted to collaborate more on philosophy. If I had some more tools here, I think I'd be more active as well. Thanks for the invitation. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:16, 19 October 2016 (UTC) ::Thanks! And thanks for creating the nomination page. I was in the process, but you beat me to it. :-) -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:01, 19 October 2016 (UTC) :::Congratulations! Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 02:47, 21 October 2016 (UTC) ::::{{Ping|Dave Braunschweig}} Definitely. Thank you again. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:19, 21 October 2016 (UTC) == Welcome == There's also {{tlx|welcomeip}}. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:25, 24 February 2017 (UTC) :{{Ping|Dave Braunschweig}} Brilliant. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:44, 24 February 2017 (UTC) == Deletion request == Hey Justin, I was wondering if you could delete [[Module:Color contrast‎]], a page I've created accidentally. I was switching between tabs with the intention of creating the page at Beta Wikiversity, and you know the rest. :) Thanks in advance. Best, [[User:Vito Genovese|{{font|color=#008000|'''Vito Genovese'''}}]] 23:10, 12 March 2017 (UTC) :{{Ping|Vito Genovese}} No problem--accidents happen. Happy to help, Vito. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:13, 12 March 2017 (UTC) == Do humans have free will? == Hi Koavf! The Wikidebate [[Do humans have free will?]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:12, 4 July 2017 (UTC) :{{Ping|Marshallsumter}} It's certainly a good start. Go for it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:14, 4 July 2017 (UTC) == Does everything happen for a sufficient reason? == Hi Koavf! [[Does everything happen for a sufficient reason?]] also appears well-developed! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:32, 4 July 2017 (UTC) :{{Ping|Marshallsumter}} Go for it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:26, 4 July 2017 (UTC) == New wikidebate syntax == Hi Justin! Just wanted to let you know that I made a new improvement to the software and syntax. It's now even cleaner and more compatible with the visual editor. Hope you like it, cheers! --[[User:Sophivorus|Felipe]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 23:58, 5 July 2017 (UTC) == Learning bass guitar with Joseph Patrick Moore == Hi Koavf! Your course [[Learning bass guitar with Joseph Patrick Moore]] appears well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:18, 19 February 2018 (UTC) :{{Ping|Marshallsumter}} Not yet, please. I'm still uploading videos and fleshing out the text portion. I'd be delighted for it to be featured soon, tho. I'll ping you when I'm done-ish. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:30, 19 February 2018 (UTC) == User:Beogradbulevar == Most posts relating to boxing or chess are from globally banned user George Reeves Person. Typical attacks come when he gets off work between 2 and 5 p.m. CST, and occasionally later, particularly on Fridays or Saturdays. He uses public libraries for Internet access, and typically doesn't post after 9 p.m. CST. It's unfortunate, but we really have to watch who posts what in the mid-to-late afternoons and be vigilant in blocking the content and not welcoming the user. See [[Wikiversity:Community Review/Marshallsumter]] for the damage it causes. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:25, 5 November 2019 (UTC) :{{Ping|Dave Braunschweig}} Wow. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:54, 5 November 2019 (UTC) == CU == I closed the CU nomination due to the low number of recent additions to the discussion. It just seemed like we wouldn't meet the criteria in a reasonable time. Thanks for offering to help with this and perhaps we can try again in the future. We appreciate your contributions. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:45, 29 January 2020 (UTC) :{{Ping|Mu301}} For sure. Thanks yourself. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:21, 29 January 2020 (UTC) == history of covid in the usa == Hi {{PAGENAME}} I was idly surfing the wsj and suddenly realized all articles I was looking at had a video posted right at the top.(example:https://www.wsj.com/articles/some-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000). The video section is 8:06 minutes long and is a short version of the history of pandemic in the usa. I don't know how to get the url of the video itself. Can you help? Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:57, 2 November 2020 (UTC) :{{Ping|Ottawahitech}} Load the page in your browser and use the networking console--you can usually get this to display by pressing F12. You'll find that this video is served up as a playlist of several bits with the URI https://oms.dowjoneson.com/b/ss/djglobal/1/JS-2.17.0/s04078897862906?AQB=1&ndh=1&pf=1&t=2%2F10%2F2020%2013%3A6%3A8%201%20300&mid=71630168209780702446627362471898499848&ce=UTF-8&pageName=WSJLive_Video_How%20Coronavirus%20Spread%20Across%20the%20U.S.%20to%20Reach%20200%2C000%20Deaths_372&g=https%3A%2F%2Fwww.wsj.com%2Farticles%2Fsome-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000&c.&a.&media.&friendlyName=How%20Coronavirus%20Spread%20Across%20the%20U.S.%20to%20Reach%20200%2C000%20Deaths&length=486&name=AE28508C-C7DF-406E-814F-69C8FAAD1A86&playerName=Web&channel=WSJ&show=Feature%20Explainer&originator=cmccall&genre=WSJ_News_U.S.%20News&digitalDate=original_2020-09-22%2011%3A58_current_2020-09-22%2011%3A58&feed=video&network=115&format=user%20initiated&streamType=video&view=true&vsid=160434036774097779839&.media&contentType=vod&.a&page.&content.&type=Article&.content&full.&url=https%3A%2F%2Fwww.wsj.com%2Farticles%2Fsome-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000&.full&site=Online%20Journal&.page&video.&player.&type=Web&technology=html%203.41.2.205&.player&keywords=CORONAVIRUS%20RESPONSE%7CCORONAVIRUS%20TESTING%7CCOVID-19%20TESTING%7CDANIELA%20HERNANDEZ%7CPANDEMIC%7CTESTING%20SITES&base.&url=https%3A%2F%2Fwww.wsj.com%2Farticles%2Fsome-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000&.base&.video&article.&id=SB11126288623532913915004586647794135594296&author=Sarah%20Toy&publish=2020-09-23%2013%3A00&publish.&orig=2020-09-23%2013%3A00&.publish&.article&ad.&blank.&start=false&.blank&disabled=true&catastrophic.&blocker=false&.catastrophic&.ad&.c&pe=ms_s&pev3=video&s=1600x900&c=24&j=1.6&v=N&k=Y&bw=781&bh=776&mcorgid=CB68E4BA55144CAA0A4C98A5%40AdobeOrg&AQE=1 or somesuch (it may not be identical for you). If you open this in VLC Player, you can save playlists as videos. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:09, 2 November 2020 (UTC) ==Custodianship== Welcome to en.wv custodianship [[User:Koavf]]. Thanks for helping. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:04, 8 September 2023 (UTC) :Merci, James. I hope I'm an asset to the community. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:50, 8 September 2023 (UTC) == Bowling article == Hey there Koavf! I've created that [[Bowling Fundamentals|bowling article]] we discussed at the Colloquium. Do you have any advice on how I can further improve it? [[User:Contributor 118,784|Contributor 118,784]] ([[User talk:Contributor 118,784|discuss]] • [[Special:Contributions/Contributor 118,784|contribs]]) 01:20, 26 September 2023 (UTC) :Nice. I don't have any particular feedback other than what I mentioned there. I'm pretty ignorant about bowling. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:26, 26 September 2023 (UTC) ::Fair, thank you! [[User:Contributor 118,784|Contributor 118,784]] ([[User talk:Contributor 118,784|discuss]] • [[Special:Contributions/Contributor 118,784|contribs]]) 09:18, 26 September 2023 (UTC) == RCA talkback (January 2024) == {{talkback|WV:RCA|User:50.118.222.66 has been flooding our abuse filter log with spam}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:31, 2 January 2024 (UTC) == Invitation to discuss page deletion policy == A discussion that might interest you has been started at [[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]. -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 15 February 2024 (UTC) == RCA talkback == {{tb|Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 24 May 2024 (UTC) == Report == Hello, I would like to report this user, who has a COI: [[Special:Contributions/Oluwadarasimi Morayo]] Thank you. [[User:Ternera|Ternera]] ([[User talk:Ternera|discuss]] • [[Special:Contributions/Ternera|contribs]]) 14:51, 24 May 2024 (UTC) :Thanks. It's best to leave these at a board like [[Wikiversity:Request custodian action]], but this was obvious spam. Cheers. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:19, 24 May 2024 (UTC) == Files == Hello! Thank you for deleting files once again! You made a comment about "all local uploads". Fair-use is not allowed on Commons so the 2,712 files in [[:Category:All non-free media]] can't go to Commons. But as I understand [[Wikiversity:Requests_for_Deletion#Deleting_ALL_non-free_uploads_by_User:Marshallsumter]] the files uploaded by Marshallsumter could be deleted. That would eliminate 1,126 files. Since [[Wikiversity:Uploading_files#Exemption_Doctrine_Policy]] allow fair use it would require a vote/discussion to change that. Young1lim uploads many pdf-files and as far as I know Commons generally do not like pdf-files. Except when it is scans of old books etc. So I do not think those files should go to Commons right now. There are still many files in [[Special:UnusedFiles]]. Right now 1,422 but some are uploaded by Young1lim. But the latest deletion request ended with delete so I think there is concensus to delete files. But some were also found good and moved to Commons. So the question is if we need another discussion about the files or if someone (you?) could just go through the files when you have a little time and either move to Commons or delete. If you think we could make one final discussion about all the files and ask for a go to the "any admin that want to can check the files and either move to Commons or delete". Then noone can come later and complain that you or another admin just deleted a file without warning. If there are 40k files in total. Perhaps 22k are pdf uploaded by Young1lim. 3k are non-free. 1.5k are unused. That would leave around 13.5k free files in use. That is a lot of files to check. I do not think there are many users that are willing to spend much time checking those files. But it would help if no more free files are uploaded (except pdf). There is allready a text on the top of [[Special:Upload]] suggesting commons. But it could perhaps be made more clear. And perhaps some of the options on [[MediaWiki:Licenses]] could be removed. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 18:01, 27 July 2024 (UTC) :Yeah, to be clear, I appreciate that sister projects like e.g. Wikibooks allow a lot of free-use files because they allow video game strategy guides and there is substantial value in screenshots or Wikipedia allows album covers and film posters as identifying media. I'm not proposing any change to policy and I accept that there are reasons for fair use, so I apologize for that sloppy wording. That said, I definitely think we should have minimal fair use if any at all. :As for PDFs, there are plenty at Commons: I have uploaded dozens and dozens of books, scientific articles, etc. It's not a problem, but it's just not optimal for many kinds of files, such as maps or something. :I'm happy to help and slog thru the uplaods if you start a conversation. Just ping me. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:49, 27 July 2024 (UTC) :: Yes fair use have some benefits. But If we/someone is going to make a cleanup it could perhaps be a good idea to first have a discussion about it. So I will start a post about fair use on wikiversity. :: And about unused files I will start a deletion discussion (again) just to be sure. :: If you feel like deleting files you could kill the files uploaded by Marshallsumter. :-) --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 09:25, 29 July 2024 (UTC) ::: I started a discussion at [[Wikiversity:Colloquium#Fair_use_on_Wikiversity]]. Lets see what happens. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 21:23, 29 July 2024 (UTC) :::: With the files of Marshallsumter gone that really helped a lot! Lets see what everyone thinks about the rest of the files. It will probably take weeks the get enough comments. But thats okay. It is summer and vacation time and if the files have been around for years they can easily wait a little longer. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:20, 31 July 2024 (UTC) Hello! Some files have been moved to Commons if you would like to have. Look 😊 --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:35, 3 March 2025 (UTC) :1,587<2,712, that's for sure. I'll try to keep chipping away at these. Thanks for the reminder. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:06, 3 March 2025 (UTC) == Revert? == Why did you revert this argument? I wanted (humorously) to make the observation that the guilty party at the end of a suicide is dead but is the only one that can be punished. Attempted and assisted suicide wasn't included. [[Special:Contributions/176.0.152.191|176.0.152.191]] ([[User talk:176.0.152.191|discuss]]) 22:27, 15 September 2024 (UTC) :It's not really a venue for hilarious jokes about killing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:44, 15 September 2024 (UTC) ::but I remember there was really some law along that line. With a similar explanation. Some king (could be from a fairy tale, but I don't believe so) wanted to outlaw suicide and his advisers had this idea. The judge (or the king himself) would speak the verdict and justice was already done. So the king was famous for his his fair and swift justice. You see I don't remember too clearly, therefore I wanted to compress the essence of this into an argument. I didn't think it was that hilarious, so sorry for injured sensitivity. Now that you know what I wanted to do, could you please formulate an accordingly compressed argument, in the appropriate tone? [[Special:Contributions/176.0.152.191|176.0.152.191]] ([[User talk:176.0.152.191|discuss]]) 00:52, 16 September 2024 (UTC) :::I think you can. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:56, 16 September 2024 (UTC) ::::I'm not a native speaker. And that you found it hilarious, where I targeted a slightly levied tone shows me that I can't really do it. [[Special:Contributions/176.0.152.191|176.0.152.191]] ([[User talk:176.0.152.191|discuss]]) 01:05, 16 September 2024 (UTC) :::::I believe in you. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:10, 16 September 2024 (UTC) == Wrong import == Hi, template:Languages does not work properly and I think its because even you states that you have importated Module which this template use from BetaWikiversity, you actually imported it from Commons, so the template is than calling non-existent function subpates. Compare: <nowiki>*</nowiki>[[Module:Languages|en.wv module Languages]] <nowiki>*</nowiki>[[commons:Module:Languages]] <nowiki>*</nowiki>[[betawikiversity:Module:Languages|betaversity]] So I dont know if removing incorect revisions and importing corect ones will fix it, but the error message is probably delivered because of this mismatch. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 13:25, 19 August 2025 (UTC) :Weird, I thought I reverted that. Let me delete that rev. So sorry. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 14:10, 19 August 2025 (UTC) == A barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[File:Resilient Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Silver Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thanks for contributing to Wikiversity for a very long time. You are the best. —[[User:RailwayEnthusiast2025|<span style="font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;">RailwayEnthusiast2025</span>]] ^{[[User talk:RailwayEnthusiast2025|<span style="color:#59a53f">''talk with me!''</span>]]} 19:55, 9 September 2025 (UTC) |} :How kind. I'm appreciate of your additions and ideas as well. Thanks so much. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:29, 9 September 2025 (UTC) == Deleting all unused templates == You seem to have been deleting many templates with the summary "unused template". One qualm I have with this is that, in general, deleting all unused templates is likely to lead to some revision histories (those that used the templates) becoming illegible. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 19 September 2025 (UTC) :Yeah, maybe. Probably not a big deal, tho. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:22, 19 September 2025 (UTC) :: In the English Wikiversity, that is plausible enough. On the other hand, in the English Wiktionary, deleting the once widely used [[wikt: T: term]] as unused would cause massive harm as for legibility, for no appreciable benefit. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:24, 19 September 2025 (UTC) :::Any examples that really matter can be undeleted or something if really necessary. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:25, 19 September 2025 (UTC) :::: I have not been long enough around the English Wikiversity to know which of the many (over 100?) deleted templates were once widely used. :::: Background: In the English Wiktionary, I noticed that someone made the thesarus revision histories completely illegible. There is too much disregard for legibility of revision histories going around. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:33, 19 September 2025 (UTC) :::::It is a concern of some regard, granted. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:44, 19 September 2025 (UTC) ::::Hi Koavf; as follow-up for this issue, I wanted to mention the [[Template:Convert links]]. This is far from being unused, since it's a fundamental tool in importing Wikipedia articles to Wikiversity, e.g. for all the Wikijournals - see step 4 of [[WikiJournal_User_Group/Editorial_guidelines#Importing_from_Wikipedia]]. ::::I just bumped into this issue myself, and I presume it will be relevant for several other users in the future. As far as I know, there are no other ways to convert those links (beside doing it manually one by one). Could you therefore please undelete that template? [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 07:56, 22 September 2025 (UTC) :::::Of course. My apologies for causing problems. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:01, 22 September 2025 (UTC) ::::::Perfect, thanks a lot! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 08:04, 22 September 2025 (UTC) I was not aware, that unused templates can be deleted without any notice. I think nothing (except obvious spam and vandalism) should be deleted without warning and time to respond.<br> [[Wikiversity:Requests_for_Deletion#Please_restore_my_templates|Please restore 61 of them.]] --[[User:Watchduck|Watchduck]] <small>([[User talk:Watchduck|quack]])</small> 15:00, 7 October 2025 (UTC) :I undeleted two templates that you asked for above, but one of them is [[Template:Studies of Euler diagrams/tamino NP table]], which is just unused. Why do these need to be here? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:01, 7 October 2025 (UTC) == Restoring Template:Copyrighted == Can you please restore [[:Template:Copyrighted]]? It is clear why this template would be unused: it is only used when some page is tagged as a possible copyright violation. I guess there should be a way to tag templates as unused-but-needed, and this would be one of then. These would then be excluded from a clean-up action like yours. On the other hand, the template is linked from [[:Wikiversity:Copyright issues]], so while it is perhaps unused in the sense of ''not invoked'', it is ''linked to''. And a clean-up should ideally not delete pages that are linked to, or consider them on a careful case-to-case basis, no? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:06, 8 October 2025 (UTC) :{{Done}} and agreed that if they have links that aren't from an old talk archive or a userspace or something more trivial, then there should at least be some appropriate action to not leave a redlink. The goal was to go back over those reports the next week or two once they've refreshed to also see wanted templates or wanted pages and try to clear those, so that two-pass system <em>should</em> catch errors like this, but not always. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:46, 8 October 2025 (UTC) == Manual numbering == My use of manual numbering in the discussion that you modified (RFD) was intentional. One can find documents using such an approach, I think. I would therefore prefer that you leave it as is next time. I am not going to revert it this time; it's not really a big deal. And thank you for correcting my misspeling of suspition to suspicion; my being a non-native speaker showed here. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:13, 9 October 2025 (UTC) :Good deal. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:15, 9 October 2025 (UTC) == Draft namespace move == Hello Justin, Do you think it is alright to move [[User:RailwayEnthusiast2025/Basic Scratch Coding]] and subpages to Draft namespace<s>.</s>? Because I <s>H</s>haven't fully completed it and would appreciate it if other contributors in the community would like to help out. Thanks, RE —[[User:RailwayEnthusiast2025|<span style="font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;">RailwayEnthusiast2025</span>]] ^{[[User talk:RailwayEnthusiast2025|<span style="color:#59a53f">''talk with me!''</span>]]} 18:27, 26 October 2025 (UTC) :I certainly think so, but honestly, I think the draft namespace is kind of a joke anyway. But I totally support you doing it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:39, 26 October 2025 (UTC) == Article Info - Related item == In the Lints was [[:User:Octfx/sandbox2]]. This was throwing a stripped Small , which I can't currently trace, Suggesting the earlier fix whilst mostly stable, has a very specfic interaction. Perhaps you can take a look and resolve this for robustness? [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 23:33, 31 October 2025 (UTC) :Diagnosing it would be optimal, but to resolve the issue, I just [https://en.wikiversity.org/w/index.php?title=User%3AOctfx%2Fsandbox2&diff=2765037&oldid=2425963 commented it out]. The page hasn't been edited in years, nor has that editor edited in years, so I just don't have the bandwidth to investigate. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:39, 31 October 2025 (UTC) == Possible copyvio == Can you please look at [[User:Harold Foppele/sandbox-2]] to see whether there is a copyvio, and if there is one, delete the page? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:45, 6 November 2025 (UTC) :@[[User:Koavf|Koavf]] Since you are a custodian, can you please put a stop to this? To me it seems like a personal vendetta that should not belong here. As for the page [[User:Harold Foppele/sandbox-2]] i asked [[user:Jtneill|Jtneill]] for advice some 12 hours ago. Since he is in Australia there is minimum a 12 hour delay in response. Would you maybe willing to help me? Kind regards, [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:58, 6 November 2025 (UTC) ::I don't know what the deal is between you and Dan, but I saw the earlier post he made to the curator's noticeboard and haven't had time to investigate. Since it seems that the two of you have some kind of friction, it may be best for you two to just generally avoid interaction in the immediate term. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:03, 6 November 2025 (UTC) :This [https://archive.org/details/Caltech-ES23.5.1960/page/2/mode/2up was published in the United States with a copyright notice, all rights reserved], so if it's in the public domain is a question of [[:c:Commons:Copyright rules by territory/United States|if the registration was renewed in a timely manner]]. Unfortunately, there is no single database of all renewals, so we can't know for sure if it <em>wasn't/t</em> renewed. We should probably err on the side of assuming that it's a copyright violation. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:02, 6 November 2025 (UTC) ::I made a request, just to make sure to:: cmgworldwide.com to obtain a license to use it in Wikiversity. As it looks for now i can get the license and will know that end next week. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:23, 6 November 2025 (UTC) :::I am going to delete it for now. It can be undeleted as necessary. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:49, 7 November 2025 (UTC) ::::👍 [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:07, 7 November 2025 (UTC) == Chess by Wikiversitans == I made a short setup for the page [[Chess/Play with other Wikiversitans]]. Is that the way you would like it to go? Do you by anychance play chess yoursef? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:21, 6 November 2025 (UTC) :Great questions. I made that page years ago and [[User:Mu301]] erroneously deleted it. I restored the old revs. As for how it should look, it's all wide open, so I have no objections. I think the notion of somehow playing here on site is actually intriguing. Maybe we could make that work... —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:05, 6 November 2025 (UTC) ::Help is needed from a specialist in the heart of Wiki. If you look at or know Lichess.org its very complex. However starting a Wikiversitans team there is a piece of cake. Just how do we invite our "members" here? Ideas welcome :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:49, 6 November 2025 (UTC) :::Would love to play chess with you. Find me at [[Chess/Play with other Wikiversitans]] in Lichess.org or Chess.com. Leave a message or email if you want to play. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:46, 7 November 2025 (UTC) ::::Thanks. I saw your invite in my inbox, but I'm a little distracted now and recently started a new job, so I didn't want to agree until I had time to actually play. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 11:48, 7 November 2025 (UTC) :::::No problem. just say "When" :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:51, 7 November 2025 (UTC) ::::::[[Chess/Board Configurations]] I think you'll like it. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 13:56, 7 November 2025 (UTC) :::::::There is also a Wikiversity chess team <span style="background-color: #aaffaa;">created at [https://lichess.org/team/wikiversity Lichess.org].</span> [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:58, 8 November 2025 (UTC) ::::::::Oh dip. Thanks for the heads up. I'm glad to see you taking initiative about this. If only I had more time myself. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:22, 8 November 2025 (UTC) == Importing template == {{Ping|Koavf}} I would like to change the [[Template:Quantum mechanics]] to look more like [[W:Template:Quantum mechanics]] since the template at WV has almost no contence I could edit that, but better ask you instead of doing it. Btw we should play chess sometime :) Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:54, 14 November 2025 (UTC) == Night mode unaware lint.. == Thanks for the edits to self. Do you plan to proceed on updating other high-use templates? like {{tl|information}}, and {{tl|article info}}, where I should ideally have resolved the Night mode unaware lint as the same time as the other fixes in the sandbox version you swapped in :(. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:42, 18 November 2025 (UTC) Please also check my contributions on talk pages for {{tl|edit protected}} requests. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:42, 18 November 2025 (UTC) :In principle, yes, I do. When will I find the time??? Note that a lot of those edit request were up for months or a year+. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:43, 18 November 2025 (UTC) : An obvious group to update would be {{tl|Projectbox}} and {{tl|Robelbox}} families, although I would strongly suggest migrating these to use template styles over the current inline approach. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:49, 18 November 2025 (UTC) ::These are good ideas, but I just don't know when I'll have time to implement them. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:40, 19 November 2025 (UTC) == Wikidebate form == Hi, hope you're doing good! I just noticed some months ago you deleted [[Template:Form/wikidebate]]. The template was indeed unused (and probably undocumented too) but it did serve a purpose, namely to be ''substituted'' when creating a new wikidebate via [[Wikidebate/New]]. As a consequence, [[Is hate is an ineffective and or selfish emotion?|this happened]] and could happen again. Could you restore it, please? If you can do that, I'll document it properly and tag it with <nowiki>__EXPECTUNUSEDTEMPLATE__</nowiki> to avoid further confusion. Thanks! [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discusión]] • [[Special:Contributions/Sophivorus|contribs.]]) 14:39, 23 December 2025 (UTC) :Of course. Thanks for your understanding. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:34, 23 December 2025 (UTC) == [[:Category:Wikiversity fully protected templates]] == I am creating semi/full protection categories for various namespace pages, so can you undelete [[:Category:Wikiversity fully protected templates]]? Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:57, 14 April 2026 (UTC) :{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:09, 14 April 2026 (UTC) == Different way to display talk pages for easier reading? == On [[Wikiversity:Colloquium]] when many people reply to the same thing all their posts are jumbled together into one big paragraph. Is this a well known problem? Is there a gadget I could use/active to make readability/accessiblity greater on Wikiversity or are we still working on that? Can I do anything obvious in order to help in this regard? ie. manually editing talk pages and adding proper wikitext or edit my own common.js? With the recent activation of a javascript that got up on the news...is there a way I can safely test my own common.js code that I ask an LLM to generate for me? I have a Qubes OS computer where I have access to disposable VMs which I can also turn off the internet on so even if the code goes haywire it won't affect my computer or the internet connection. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 11:45, 27 April 2026 (UTC) azpyeym0jcf8vxie4abuk9qslf95hb3 2806783 2806782 2026-04-27T11:49:09Z ThinkingScience 3061446 /* Different way to display talk pages for easier reading? */ two 'major' spelling issues my LLM missed 2806783 wikitext text/x-wiki {| style="border-spacing:8px;margin:0px -8px" width="100%" |class="MainPageBG" style="width: 55%; border:1px solid #084080; background-color:#F5FFFA; vertical-align:top;color:#000000;font-size: 85%"| {| width="100%" cellpadding="2" cellspacing="5" style="vertical-align:top; background-color:#F5FFFA" ! <div style="margin: 0; background-color:#CEF2E0; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #084080; text-align:left; color:#082840; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;"> '''Hello Koavf! [[Wikiversity:Welcome, newcomers|Welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you decide that you need help, check out [[Wikiversity:Help desk]], ask the [[Wikiversity:Support staff|support staff]], or ask me on my talk page. Please remember to [[Wikiversity:Sign your posts on talk pages|sign your name]] on talk pages using four tildes (~&#126;~~); this will automatically produce your name and the date. Below are some recommended guidelines to facilitate your involvement. Happy Editing! -- [[User:Trevor MacInnis|Trevor MacInnis]] 22:28, 4 September 2006 (UTC)</div> |} {| style="border-spacing:8px;margin:0px -8px" width="100%" |class="MainPageBG" style="width: 55%; border:1px solid #FFFFFF; background-color:#F5FFFA; vertical-align:top"| {| width="100%" cellpadding="2" cellspacing="5" style="vertical-align:top; background-color:#F5FFFA" ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #CEF2E0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting Started</div> |- |style="color:#000"| * [[Wikiversity:Guided tour|Take a guided tour]] * [[Help:Editing|How to edit a page]] * [[Wikiversity:Be bold|Be bold in editing]] * [[Portal:Learning Projects|Learning Projects]] * [[Wikiversity:What Wikiversity is not|What Wikiversity is not]] |- ! <div style="margin: 0; background:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting your info out there</div> |- | style="color:#000"| * [[Wikiversity:Cite sources|Cite your sources]] * [[Wikiversity:Disclosures|Neutral Point of View]] * [[Wikiversity:Verifiability|Verifiability]] |- ! <div style="margin: 0; background:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting more Wikiversity rules</div> |- | style="color:#000"| * [[Wikiversity:Policies|Policy Library]] |- |} |class="MainPageBG" style="width: 55%; border:1px solid #FFFFFF; background-color:#F5FFFA; vertical-align:top"| {| width="100%" cellpadding="2" cellspacing="5" style="vertical-align:top; background-color:#F5FFFA" ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #CEF2E0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting Help</div> |- |style="color:#000"| * [[Wikiversity:Research|Research guidelines]] * [[Wikiversity:Help desk|Help Desk]] |- ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting along</div> |- |style="color:#000"| * [[Wikiversity:Civility|Civility]] * [[Wikiversity:Sign your posts on talk pages|Sign your posts]] * [[Wikiversity:Scholarly ethics|Scholarly ethics]] |- ! <div style="margin: 0; background-color:#084080; font-family: sans-serif; font-size:120%; font-weight:bold; border:1px solid #cef2e0; text-align:left; color:#FFC000; padding-left:0.4em; padding-top: 0.2em; padding-bottom: 0.2em;">Getting technical</div> |- |style="color:#000"| [[Image:Wikimedia Foundation RGB logo with text.svg|60px|right]] * [[Wikiversity:Colloquium|Colloquium]] |- |} |} |} == wikitravel == Hi. You removed links to Wikitravel. Why? --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 12:44, 24 October 2013 (UTC) :'''Wikitravel links''' Per discussion at [[w:Template:Wikitravel|en.wp]] as well as [[m:Interwiki map|Meta]] to remove links at those projects. If you want to keep links and references here at en.v, I guess that's fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:28, 24 October 2013 (UTC) == Thanks. == I see you got it before I explained. Wikiversity is disconcerting to those familiar with the encyclopedia projects, and the other content-oriented projects. While we do have a content mission, we ''also'' have a "learning by doing" mission, which is about ''people.'' Our product is not just content, it is education, and there is no education without users who are educated, and sophisticated education is always about process and people skills and the rest. I would argue that the encyclopedia projects also need to be welcoming, if the full mission is to be fulfilled, but ... they developed with a very narrow focus and absent the realization that an environment that was easily seen as hostile would damage the mission. The 20th century saw the development of systems and skills and process for maximizing consensus, and the only reliable measure of neutrality is level of consensus. (I.e., if everyone involved agrees, 100% consensus, while what they agree upon only might possibly turn out, in the end, to be defective or invalid, there is no better measure!). So to the extent that there is exclusion, to that extent, the assessment of neutrality can be warped. Obviously, compromises are necessary, but "compromise" requires tolerating a level of damage, and that is easily forgotten. When the importance of consensus being as broad as possible is realized, a community will find ways to keep conversation open, on some level, in some place, otherwise the community becomes locked into what I call the "tyranny of the past." There is a children's song that was part of a therapeutic response to Reactive Attachment Disorder: :'''There is always something you can do, do, do''' :'''When you're getting in a stew, stew, stew.''' Mostly, it involves simmering down, dropping upset and reactive response, and, when calm, communicating. While this kind of work has been done on Wikipedia, often in user space -- it's what I did, successfully mediating disputes, such that users at each other's throats became cooperative ''with each other'' -- this was mumbo-jumbo nonsense to too many on Wikipedia. For example, see [https://en.wikipedia.org/wiki/Wikipedia:Miscellany_for_deletion/Abd_user_pages], which included many pages of historical function, including evidence presented to ArbCom. I found it very strange that ArbCom did not care that evidence used in a case was being deleted, but ArbCom consists of too many elevated beyond their competence by popularity (as well as many other highly-experienced and thoughtful user; but the system tends to burn them out and they become less attentive.) [[w:User:Abd/Dispute over thermoeconomics]] was particularly educational. In that mediation, a professor was revert warring with Randy from Boise, so to speak, and one or both were about to be blocked. It took very little to develop cooperation, mostly just sitting them down together with some support. Hmmm... I'm thinking of asking that these pages be transwikied to Wikiversity, precisely for historical study. Looking for the link to that, I came across [https://en.wikipedia.org/wiki/Wikipedia:Miscellany_for_deletion/User:UBX/Esperanza_returns this]. It shows a quick and major clue to what happened on en.wiki. Two three-letter users with a conflict. One was an administrator taken to ArbCom by the other, and the administrator was trout-slapped by ArbCom and then, it is obvious, revenge was exacted, by the admin and his friends. This was long-continued and, while not unnoticed, never sanctioned. Admins can be hostile, this one was more than hostile, he was highly insulting at times, using obscene language, and using tools while involved, was reprimanded, made small adjustments to his behavior, but continued pretty much unimpeded. And, as you know, this is not uncommon. He is even a likeable Guy. I consider this all the responsibility ''of the community.'' Blaming people for what comes naturally for them is not productive. Such people generally will modify behavior in a functional community. Notice the irony. The userbox was "Esperanza returns," referring to the project designed to foster civility and welcome and cooperation. Esperanza, of course, means Hope. So the nominator was saying, "Hope will never return." Esperanza was crushed when it temporarily was inactive. Instead of improving the governance, which was easily possible, it was crushed with ''vehemence,'' see the [[w:Wikipedia:Miscellany_for_deletion/Wikipedia:Esperanza|MfD]]. Why? To any serious student of human organizational structure, it's obvious. Wikiversity is the slim thread of hope, and if it is not protected and defended, hope will break. Thanks again. --[[User:Abd|Abd]] ([[User talk:Abd|discuss]] • [[Special:Contributions/Abd|contribs]]) 15:17, 7 August 2015 (UTC) == Curator == Hi! I've noticed and appreciated your recent efforts on behalf of Wikiversity. Do you have any interest in becoming a [[Wikiversity:Curators|Wikiversity curator]]? It would give you additional tools to make some clean-up easier. I'd be happy to nominate/support you if you are interested. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:11, 19 October 2016 (UTC) :{{Ping|Dave Braunschweig}} I'd be delited and honored. I started editing here as soon as it was founded and I've always wanted to collaborate more on philosophy. If I had some more tools here, I think I'd be more active as well. Thanks for the invitation. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:16, 19 October 2016 (UTC) ::Thanks! And thanks for creating the nomination page. I was in the process, but you beat me to it. :-) -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:01, 19 October 2016 (UTC) :::Congratulations! Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 02:47, 21 October 2016 (UTC) ::::{{Ping|Dave Braunschweig}} Definitely. Thank you again. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:19, 21 October 2016 (UTC) == Welcome == There's also {{tlx|welcomeip}}. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:25, 24 February 2017 (UTC) :{{Ping|Dave Braunschweig}} Brilliant. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:44, 24 February 2017 (UTC) == Deletion request == Hey Justin, I was wondering if you could delete [[Module:Color contrast‎]], a page I've created accidentally. I was switching between tabs with the intention of creating the page at Beta Wikiversity, and you know the rest. :) Thanks in advance. Best, [[User:Vito Genovese|{{font|color=#008000|'''Vito Genovese'''}}]] 23:10, 12 March 2017 (UTC) :{{Ping|Vito Genovese}} No problem--accidents happen. Happy to help, Vito. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:13, 12 March 2017 (UTC) == Do humans have free will? == Hi Koavf! The Wikidebate [[Do humans have free will?]] appears to be well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:12, 4 July 2017 (UTC) :{{Ping|Marshallsumter}} It's certainly a good start. Go for it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:14, 4 July 2017 (UTC) == Does everything happen for a sufficient reason? == Hi Koavf! [[Does everything happen for a sufficient reason?]] also appears well-developed! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:32, 4 July 2017 (UTC) :{{Ping|Marshallsumter}} Go for it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:26, 4 July 2017 (UTC) == New wikidebate syntax == Hi Justin! Just wanted to let you know that I made a new improvement to the software and syntax. It's now even cleaner and more compatible with the visual editor. Hope you like it, cheers! --[[User:Sophivorus|Felipe]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 23:58, 5 July 2017 (UTC) == Learning bass guitar with Joseph Patrick Moore == Hi Koavf! Your course [[Learning bass guitar with Joseph Patrick Moore]] appears well-developed and ready for learners! Would you like to have it announced on our Main Page News? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:18, 19 February 2018 (UTC) :{{Ping|Marshallsumter}} Not yet, please. I'm still uploading videos and fleshing out the text portion. I'd be delighted for it to be featured soon, tho. I'll ping you when I'm done-ish. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:30, 19 February 2018 (UTC) == User:Beogradbulevar == Most posts relating to boxing or chess are from globally banned user George Reeves Person. Typical attacks come when he gets off work between 2 and 5 p.m. CST, and occasionally later, particularly on Fridays or Saturdays. He uses public libraries for Internet access, and typically doesn't post after 9 p.m. CST. It's unfortunate, but we really have to watch who posts what in the mid-to-late afternoons and be vigilant in blocking the content and not welcoming the user. See [[Wikiversity:Community Review/Marshallsumter]] for the damage it causes. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:25, 5 November 2019 (UTC) :{{Ping|Dave Braunschweig}} Wow. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:54, 5 November 2019 (UTC) == CU == I closed the CU nomination due to the low number of recent additions to the discussion. It just seemed like we wouldn't meet the criteria in a reasonable time. Thanks for offering to help with this and perhaps we can try again in the future. We appreciate your contributions. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:45, 29 January 2020 (UTC) :{{Ping|Mu301}} For sure. Thanks yourself. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:21, 29 January 2020 (UTC) == history of covid in the usa == Hi {{PAGENAME}} I was idly surfing the wsj and suddenly realized all articles I was looking at had a video posted right at the top.(example:https://www.wsj.com/articles/some-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000). The video section is 8:06 minutes long and is a short version of the history of pandemic in the usa. I don't know how to get the url of the video itself. Can you help? Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:57, 2 November 2020 (UTC) :{{Ping|Ottawahitech}} Load the page in your browser and use the networking console--you can usually get this to display by pressing F12. You'll find that this video is served up as a playlist of several bits with the URI https://oms.dowjoneson.com/b/ss/djglobal/1/JS-2.17.0/s04078897862906?AQB=1&ndh=1&pf=1&t=2%2F10%2F2020%2013%3A6%3A8%201%20300&mid=71630168209780702446627362471898499848&ce=UTF-8&pageName=WSJLive_Video_How%20Coronavirus%20Spread%20Across%20the%20U.S.%20to%20Reach%20200%2C000%20Deaths_372&g=https%3A%2F%2Fwww.wsj.com%2Farticles%2Fsome-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000&c.&a.&media.&friendlyName=How%20Coronavirus%20Spread%20Across%20the%20U.S.%20to%20Reach%20200%2C000%20Deaths&length=486&name=AE28508C-C7DF-406E-814F-69C8FAAD1A86&playerName=Web&channel=WSJ&show=Feature%20Explainer&originator=cmccall&genre=WSJ_News_U.S.%20News&digitalDate=original_2020-09-22%2011%3A58_current_2020-09-22%2011%3A58&feed=video&network=115&format=user%20initiated&streamType=video&view=true&vsid=160434036774097779839&.media&contentType=vod&.a&page.&content.&type=Article&.content&full.&url=https%3A%2F%2Fwww.wsj.com%2Farticles%2Fsome-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000&.full&site=Online%20Journal&.page&video.&player.&type=Web&technology=html%203.41.2.205&.player&keywords=CORONAVIRUS%20RESPONSE%7CCORONAVIRUS%20TESTING%7CCOVID-19%20TESTING%7CDANIELA%20HERNANDEZ%7CPANDEMIC%7CTESTING%20SITES&base.&url=https%3A%2F%2Fwww.wsj.com%2Farticles%2Fsome-covid-19-patients-show-signs-of-heart-damage-months-later-11600866000&.base&.video&article.&id=SB11126288623532913915004586647794135594296&author=Sarah%20Toy&publish=2020-09-23%2013%3A00&publish.&orig=2020-09-23%2013%3A00&.publish&.article&ad.&blank.&start=false&.blank&disabled=true&catastrophic.&blocker=false&.catastrophic&.ad&.c&pe=ms_s&pev3=video&s=1600x900&c=24&j=1.6&v=N&k=Y&bw=781&bh=776&mcorgid=CB68E4BA55144CAA0A4C98A5%40AdobeOrg&AQE=1 or somesuch (it may not be identical for you). If you open this in VLC Player, you can save playlists as videos. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:09, 2 November 2020 (UTC) ==Custodianship== Welcome to en.wv custodianship [[User:Koavf]]. Thanks for helping. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:04, 8 September 2023 (UTC) :Merci, James. I hope I'm an asset to the community. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:50, 8 September 2023 (UTC) == Bowling article == Hey there Koavf! I've created that [[Bowling Fundamentals|bowling article]] we discussed at the Colloquium. Do you have any advice on how I can further improve it? [[User:Contributor 118,784|Contributor 118,784]] ([[User talk:Contributor 118,784|discuss]] • [[Special:Contributions/Contributor 118,784|contribs]]) 01:20, 26 September 2023 (UTC) :Nice. I don't have any particular feedback other than what I mentioned there. I'm pretty ignorant about bowling. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:26, 26 September 2023 (UTC) ::Fair, thank you! [[User:Contributor 118,784|Contributor 118,784]] ([[User talk:Contributor 118,784|discuss]] • [[Special:Contributions/Contributor 118,784|contribs]]) 09:18, 26 September 2023 (UTC) == RCA talkback (January 2024) == {{talkback|WV:RCA|User:50.118.222.66 has been flooding our abuse filter log with spam}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:31, 2 January 2024 (UTC) == Invitation to discuss page deletion policy == A discussion that might interest you has been started at [[Wikiversity:Requests_for_Deletion#Wikiversity:Deletion_Convention_2024]]. -- [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 15 February 2024 (UTC) == RCA talkback == {{tb|Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 24 May 2024 (UTC) == Report == Hello, I would like to report this user, who has a COI: [[Special:Contributions/Oluwadarasimi Morayo]] Thank you. [[User:Ternera|Ternera]] ([[User talk:Ternera|discuss]] • [[Special:Contributions/Ternera|contribs]]) 14:51, 24 May 2024 (UTC) :Thanks. It's best to leave these at a board like [[Wikiversity:Request custodian action]], but this was obvious spam. Cheers. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:19, 24 May 2024 (UTC) == Files == Hello! Thank you for deleting files once again! You made a comment about "all local uploads". Fair-use is not allowed on Commons so the 2,712 files in [[:Category:All non-free media]] can't go to Commons. But as I understand [[Wikiversity:Requests_for_Deletion#Deleting_ALL_non-free_uploads_by_User:Marshallsumter]] the files uploaded by Marshallsumter could be deleted. That would eliminate 1,126 files. Since [[Wikiversity:Uploading_files#Exemption_Doctrine_Policy]] allow fair use it would require a vote/discussion to change that. Young1lim uploads many pdf-files and as far as I know Commons generally do not like pdf-files. Except when it is scans of old books etc. So I do not think those files should go to Commons right now. There are still many files in [[Special:UnusedFiles]]. Right now 1,422 but some are uploaded by Young1lim. But the latest deletion request ended with delete so I think there is concensus to delete files. But some were also found good and moved to Commons. So the question is if we need another discussion about the files or if someone (you?) could just go through the files when you have a little time and either move to Commons or delete. If you think we could make one final discussion about all the files and ask for a go to the "any admin that want to can check the files and either move to Commons or delete". Then noone can come later and complain that you or another admin just deleted a file without warning. If there are 40k files in total. Perhaps 22k are pdf uploaded by Young1lim. 3k are non-free. 1.5k are unused. That would leave around 13.5k free files in use. That is a lot of files to check. I do not think there are many users that are willing to spend much time checking those files. But it would help if no more free files are uploaded (except pdf). There is allready a text on the top of [[Special:Upload]] suggesting commons. But it could perhaps be made more clear. And perhaps some of the options on [[MediaWiki:Licenses]] could be removed. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 18:01, 27 July 2024 (UTC) :Yeah, to be clear, I appreciate that sister projects like e.g. Wikibooks allow a lot of free-use files because they allow video game strategy guides and there is substantial value in screenshots or Wikipedia allows album covers and film posters as identifying media. I'm not proposing any change to policy and I accept that there are reasons for fair use, so I apologize for that sloppy wording. That said, I definitely think we should have minimal fair use if any at all. :As for PDFs, there are plenty at Commons: I have uploaded dozens and dozens of books, scientific articles, etc. It's not a problem, but it's just not optimal for many kinds of files, such as maps or something. :I'm happy to help and slog thru the uplaods if you start a conversation. Just ping me. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:49, 27 July 2024 (UTC) :: Yes fair use have some benefits. But If we/someone is going to make a cleanup it could perhaps be a good idea to first have a discussion about it. So I will start a post about fair use on wikiversity. :: And about unused files I will start a deletion discussion (again) just to be sure. :: If you feel like deleting files you could kill the files uploaded by Marshallsumter. :-) --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 09:25, 29 July 2024 (UTC) ::: I started a discussion at [[Wikiversity:Colloquium#Fair_use_on_Wikiversity]]. Lets see what happens. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 21:23, 29 July 2024 (UTC) :::: With the files of Marshallsumter gone that really helped a lot! Lets see what everyone thinks about the rest of the files. It will probably take weeks the get enough comments. But thats okay. It is summer and vacation time and if the files have been around for years they can easily wait a little longer. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:20, 31 July 2024 (UTC) Hello! Some files have been moved to Commons if you would like to have. Look 😊 --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:35, 3 March 2025 (UTC) :1,587<2,712, that's for sure. I'll try to keep chipping away at these. Thanks for the reminder. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:06, 3 March 2025 (UTC) == Revert? == Why did you revert this argument? I wanted (humorously) to make the observation that the guilty party at the end of a suicide is dead but is the only one that can be punished. Attempted and assisted suicide wasn't included. [[Special:Contributions/176.0.152.191|176.0.152.191]] ([[User talk:176.0.152.191|discuss]]) 22:27, 15 September 2024 (UTC) :It's not really a venue for hilarious jokes about killing. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:44, 15 September 2024 (UTC) ::but I remember there was really some law along that line. With a similar explanation. Some king (could be from a fairy tale, but I don't believe so) wanted to outlaw suicide and his advisers had this idea. The judge (or the king himself) would speak the verdict and justice was already done. So the king was famous for his his fair and swift justice. You see I don't remember too clearly, therefore I wanted to compress the essence of this into an argument. I didn't think it was that hilarious, so sorry for injured sensitivity. Now that you know what I wanted to do, could you please formulate an accordingly compressed argument, in the appropriate tone? [[Special:Contributions/176.0.152.191|176.0.152.191]] ([[User talk:176.0.152.191|discuss]]) 00:52, 16 September 2024 (UTC) :::I think you can. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:56, 16 September 2024 (UTC) ::::I'm not a native speaker. And that you found it hilarious, where I targeted a slightly levied tone shows me that I can't really do it. [[Special:Contributions/176.0.152.191|176.0.152.191]] ([[User talk:176.0.152.191|discuss]]) 01:05, 16 September 2024 (UTC) :::::I believe in you. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 01:10, 16 September 2024 (UTC) == Wrong import == Hi, template:Languages does not work properly and I think its because even you states that you have importated Module which this template use from BetaWikiversity, you actually imported it from Commons, so the template is than calling non-existent function subpates. Compare: <nowiki>*</nowiki>[[Module:Languages|en.wv module Languages]] <nowiki>*</nowiki>[[commons:Module:Languages]] <nowiki>*</nowiki>[[betawikiversity:Module:Languages|betaversity]] So I dont know if removing incorect revisions and importing corect ones will fix it, but the error message is probably delivered because of this mismatch. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 13:25, 19 August 2025 (UTC) :Weird, I thought I reverted that. Let me delete that rev. So sorry. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 14:10, 19 August 2025 (UTC) == A barnstar for you! == {| style="border: 1px solid gray; background-color: #ffffff;" |rowspan="2" valign="middle" | [[File:Resilient Barnstar.png|100px]] |rowspan="2" | |style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;" | '''The Silver Barnstar''' |- |style="vertical-align: middle; border-top: 1px solid gray;" | Thanks for contributing to Wikiversity for a very long time. You are the best. —[[User:RailwayEnthusiast2025|<span style="font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;">RailwayEnthusiast2025</span>]] ^{[[User talk:RailwayEnthusiast2025|<span style="color:#59a53f">''talk with me!''</span>]]} 19:55, 9 September 2025 (UTC) |} :How kind. I'm appreciate of your additions and ideas as well. Thanks so much. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:29, 9 September 2025 (UTC) == Deleting all unused templates == You seem to have been deleting many templates with the summary "unused template". One qualm I have with this is that, in general, deleting all unused templates is likely to lead to some revision histories (those that used the templates) becoming illegible. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:21, 19 September 2025 (UTC) :Yeah, maybe. Probably not a big deal, tho. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:22, 19 September 2025 (UTC) :: In the English Wikiversity, that is plausible enough. On the other hand, in the English Wiktionary, deleting the once widely used [[wikt: T: term]] as unused would cause massive harm as for legibility, for no appreciable benefit. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:24, 19 September 2025 (UTC) :::Any examples that really matter can be undeleted or something if really necessary. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:25, 19 September 2025 (UTC) :::: I have not been long enough around the English Wikiversity to know which of the many (over 100?) deleted templates were once widely used. :::: Background: In the English Wiktionary, I noticed that someone made the thesarus revision histories completely illegible. There is too much disregard for legibility of revision histories going around. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:33, 19 September 2025 (UTC) :::::It is a concern of some regard, granted. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:44, 19 September 2025 (UTC) ::::Hi Koavf; as follow-up for this issue, I wanted to mention the [[Template:Convert links]]. This is far from being unused, since it's a fundamental tool in importing Wikipedia articles to Wikiversity, e.g. for all the Wikijournals - see step 4 of [[WikiJournal_User_Group/Editorial_guidelines#Importing_from_Wikipedia]]. ::::I just bumped into this issue myself, and I presume it will be relevant for several other users in the future. As far as I know, there are no other ways to convert those links (beside doing it manually one by one). Could you therefore please undelete that template? [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 07:56, 22 September 2025 (UTC) :::::Of course. My apologies for causing problems. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:01, 22 September 2025 (UTC) ::::::Perfect, thanks a lot! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 08:04, 22 September 2025 (UTC) I was not aware, that unused templates can be deleted without any notice. I think nothing (except obvious spam and vandalism) should be deleted without warning and time to respond.<br> [[Wikiversity:Requests_for_Deletion#Please_restore_my_templates|Please restore 61 of them.]] --[[User:Watchduck|Watchduck]] <small>([[User talk:Watchduck|quack]])</small> 15:00, 7 October 2025 (UTC) :I undeleted two templates that you asked for above, but one of them is [[Template:Studies of Euler diagrams/tamino NP table]], which is just unused. Why do these need to be here? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:01, 7 October 2025 (UTC) == Restoring Template:Copyrighted == Can you please restore [[:Template:Copyrighted]]? It is clear why this template would be unused: it is only used when some page is tagged as a possible copyright violation. I guess there should be a way to tag templates as unused-but-needed, and this would be one of then. These would then be excluded from a clean-up action like yours. On the other hand, the template is linked from [[:Wikiversity:Copyright issues]], so while it is perhaps unused in the sense of ''not invoked'', it is ''linked to''. And a clean-up should ideally not delete pages that are linked to, or consider them on a careful case-to-case basis, no? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 04:06, 8 October 2025 (UTC) :{{Done}} and agreed that if they have links that aren't from an old talk archive or a userspace or something more trivial, then there should at least be some appropriate action to not leave a redlink. The goal was to go back over those reports the next week or two once they've refreshed to also see wanted templates or wanted pages and try to clear those, so that two-pass system <em>should</em> catch errors like this, but not always. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:46, 8 October 2025 (UTC) == Manual numbering == My use of manual numbering in the discussion that you modified (RFD) was intentional. One can find documents using such an approach, I think. I would therefore prefer that you leave it as is next time. I am not going to revert it this time; it's not really a big deal. And thank you for correcting my misspeling of suspition to suspicion; my being a non-native speaker showed here. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:13, 9 October 2025 (UTC) :Good deal. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:15, 9 October 2025 (UTC) == Draft namespace move == Hello Justin, Do you think it is alright to move [[User:RailwayEnthusiast2025/Basic Scratch Coding]] and subpages to Draft namespace<s>.</s>? Because I <s>H</s>haven't fully completed it and would appreciate it if other contributors in the community would like to help out. Thanks, RE —[[User:RailwayEnthusiast2025|<span style="font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;">RailwayEnthusiast2025</span>]] ^{[[User talk:RailwayEnthusiast2025|<span style="color:#59a53f">''talk with me!''</span>]]} 18:27, 26 October 2025 (UTC) :I certainly think so, but honestly, I think the draft namespace is kind of a joke anyway. But I totally support you doing it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:39, 26 October 2025 (UTC) == Article Info - Related item == In the Lints was [[:User:Octfx/sandbox2]]. This was throwing a stripped Small , which I can't currently trace, Suggesting the earlier fix whilst mostly stable, has a very specfic interaction. Perhaps you can take a look and resolve this for robustness? [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 23:33, 31 October 2025 (UTC) :Diagnosing it would be optimal, but to resolve the issue, I just [https://en.wikiversity.org/w/index.php?title=User%3AOctfx%2Fsandbox2&diff=2765037&oldid=2425963 commented it out]. The page hasn't been edited in years, nor has that editor edited in years, so I just don't have the bandwidth to investigate. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:39, 31 October 2025 (UTC) == Possible copyvio == Can you please look at [[User:Harold Foppele/sandbox-2]] to see whether there is a copyvio, and if there is one, delete the page? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 18:45, 6 November 2025 (UTC) :@[[User:Koavf|Koavf]] Since you are a custodian, can you please put a stop to this? To me it seems like a personal vendetta that should not belong here. As for the page [[User:Harold Foppele/sandbox-2]] i asked [[user:Jtneill|Jtneill]] for advice some 12 hours ago. Since he is in Australia there is minimum a 12 hour delay in response. Would you maybe willing to help me? Kind regards, [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:58, 6 November 2025 (UTC) ::I don't know what the deal is between you and Dan, but I saw the earlier post he made to the curator's noticeboard and haven't had time to investigate. Since it seems that the two of you have some kind of friction, it may be best for you two to just generally avoid interaction in the immediate term. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:03, 6 November 2025 (UTC) :This [https://archive.org/details/Caltech-ES23.5.1960/page/2/mode/2up was published in the United States with a copyright notice, all rights reserved], so if it's in the public domain is a question of [[:c:Commons:Copyright rules by territory/United States|if the registration was renewed in a timely manner]]. Unfortunately, there is no single database of all renewals, so we can't know for sure if it <em>wasn't/t</em> renewed. We should probably err on the side of assuming that it's a copyright violation. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:02, 6 November 2025 (UTC) ::I made a request, just to make sure to:: cmgworldwide.com to obtain a license to use it in Wikiversity. As it looks for now i can get the license and will know that end next week. Thanks [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:23, 6 November 2025 (UTC) :::I am going to delete it for now. It can be undeleted as necessary. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:49, 7 November 2025 (UTC) ::::👍 [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 09:07, 7 November 2025 (UTC) == Chess by Wikiversitans == I made a short setup for the page [[Chess/Play with other Wikiversitans]]. Is that the way you would like it to go? Do you by anychance play chess yoursef? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:21, 6 November 2025 (UTC) :Great questions. I made that page years ago and [[User:Mu301]] erroneously deleted it. I restored the old revs. As for how it should look, it's all wide open, so I have no objections. I think the notion of somehow playing here on site is actually intriguing. Maybe we could make that work... —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:05, 6 November 2025 (UTC) ::Help is needed from a specialist in the heart of Wiki. If you look at or know Lichess.org its very complex. However starting a Wikiversitans team there is a piece of cake. Just how do we invite our "members" here? Ideas welcome :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:49, 6 November 2025 (UTC) :::Would love to play chess with you. Find me at [[Chess/Play with other Wikiversitans]] in Lichess.org or Chess.com. Leave a message or email if you want to play. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:46, 7 November 2025 (UTC) ::::Thanks. I saw your invite in my inbox, but I'm a little distracted now and recently started a new job, so I didn't want to agree until I had time to actually play. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 11:48, 7 November 2025 (UTC) :::::No problem. just say "When" :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:51, 7 November 2025 (UTC) ::::::[[Chess/Board Configurations]] I think you'll like it. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 13:56, 7 November 2025 (UTC) :::::::There is also a Wikiversity chess team <span style="background-color: #aaffaa;">created at [https://lichess.org/team/wikiversity Lichess.org].</span> [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:58, 8 November 2025 (UTC) ::::::::Oh dip. Thanks for the heads up. I'm glad to see you taking initiative about this. If only I had more time myself. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:22, 8 November 2025 (UTC) == Importing template == {{Ping|Koavf}} I would like to change the [[Template:Quantum mechanics]] to look more like [[W:Template:Quantum mechanics]] since the template at WV has almost no contence I could edit that, but better ask you instead of doing it. Btw we should play chess sometime :) Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:54, 14 November 2025 (UTC) == Night mode unaware lint.. == Thanks for the edits to self. Do you plan to proceed on updating other high-use templates? like {{tl|information}}, and {{tl|article info}}, where I should ideally have resolved the Night mode unaware lint as the same time as the other fixes in the sandbox version you swapped in :(. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:42, 18 November 2025 (UTC) Please also check my contributions on talk pages for {{tl|edit protected}} requests. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:42, 18 November 2025 (UTC) :In principle, yes, I do. When will I find the time??? Note that a lot of those edit request were up for months or a year+. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:43, 18 November 2025 (UTC) : An obvious group to update would be {{tl|Projectbox}} and {{tl|Robelbox}} families, although I would strongly suggest migrating these to use template styles over the current inline approach. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:49, 18 November 2025 (UTC) ::These are good ideas, but I just don't know when I'll have time to implement them. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:40, 19 November 2025 (UTC) == Wikidebate form == Hi, hope you're doing good! I just noticed some months ago you deleted [[Template:Form/wikidebate]]. The template was indeed unused (and probably undocumented too) but it did serve a purpose, namely to be ''substituted'' when creating a new wikidebate via [[Wikidebate/New]]. As a consequence, [[Is hate is an ineffective and or selfish emotion?|this happened]] and could happen again. Could you restore it, please? If you can do that, I'll document it properly and tag it with <nowiki>__EXPECTUNUSEDTEMPLATE__</nowiki> to avoid further confusion. Thanks! [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discusión]] • [[Special:Contributions/Sophivorus|contribs.]]) 14:39, 23 December 2025 (UTC) :Of course. Thanks for your understanding. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:34, 23 December 2025 (UTC) == [[:Category:Wikiversity fully protected templates]] == I am creating semi/full protection categories for various namespace pages, so can you undelete [[:Category:Wikiversity fully protected templates]]? Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:57, 14 April 2026 (UTC) :{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:09, 14 April 2026 (UTC) == Different way to display talk pages for easier reading? == On [[Wikiversity:Colloquium]] when many people reply to the same thing all their posts are jumbled together into one big paragraph. Is this a well known problem? Is there a gadget I could use/activate to make readability/accessibility greater on Wikiversity or are we still working on that? Can I do anything obvious in order to help in this regard? ie. manually editing talk pages and adding proper wikitext or edit my own common.js? With the recent activation of a javascript that got up on the news...is there a way I can safely test my own common.js code that I ask an LLM to generate for me? I have a Qubes OS computer where I have access to disposable VMs which I can also turn off the internet on so even if the code goes haywire it won't affect my computer or the internet connection. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 11:45, 27 April 2026 (UTC) o0smtsmaqkd2lba7ue8mr6oiskezs10 0 35313 2806749 2762720 2026-04-27T04:12:18Z CommonsDelinker 9184 Removing [[:c:File:Copper_Brahma_Airport.jpg|Copper_Brahma_Airport.jpg]], it has been deleted from Commons by [[:c:User:Abzeronow|Abzeronow]] because: per [[:c:Commons:Deletion requests/Files in Category:Siem Reap-Angkor International Airport|]]. 2806749 wikitext text/x-wiki '''Hinduism''' is the predominant and indigenous religious tradition of the Indian subcontinent. Hinduism is often referred to as Sanātana Dharma (a Sanskrit phrase meaning "the eternal law") by its adherents. Generic "types" of Hinduism that attempt to accommodate a variety of complex views span folk and Vedic Hinduism to bhakti tradition, as in Vaishnavism. Hinduism also includes yogic traditions and a wide spectrum of "daily morality" based on the notion of karma and societal norms such as Hindu marriage customs. [[File:Shiva Statue Murdeshwara Temple.jpg|thumb|Shiva]] [[File:Jai Hanuman.jpg|thumb|Hanuman]] [[File:Durga mata statue in Vrindavan.jpg|thumb|Durga]] [[File:Sai Baba statue.jpg|thumb|Sai Baba of Shirdi]] [[File:Sri Mariamman Temple Singapore 2 amk.jpg|thumb|Krishna]] [[File:Annamayya at Rajampet, Kadapa District.JPG|thumb|Annamacharya]] [[File:Gombak_Selangor_Batu-Caves-01.jpg|thumb|Murugan]] [[File:GWK_Cultural_Park_in_Bali.jpg|thumb|Vishnu]] [[File:Basava Gaint Statue 108 feet, Basava Kalyana.JPG|thumb|Basava]] Hinduism is formed of diverse traditions and has no single founder. Among its roots is the historical Vedic religion of Iron Age India, and as such Hinduism is often called the "oldest living religion" or the "oldest living major tradition". Demographically, Hinduism is the world's third largest religion, after Christianity and Islam, with more than a billion adherents, of whom approximately 1 billion live in the Republic of India. Other significant populations are found in Nepal (23 million), Bangladesh (14 million) and the Indonesian island of Bali (3.3 million). A large body of texts is classified as Hindu, divided into Śruti ("revealed") and Smriti ("remembered") texts. These texts discuss theology, philosophy and mythology, and provide information on the practice of dharma (religious living). Among these texts, the Vedas are the foremost in authority, importance and antiquity. Other major scriptures include the Upanishads, Purāṇas and the epics Mahābhārata and Rāmāyaṇa. The Bhagavad Gītā, a treatise from the Mahābhārata, spoken by Krishna, is of special importance. ==Resources== {{wikibooks|God and Religious Toleration/Hinduism}} * [[/Books/]] * [[/Bhagavad Gita/]] ==Sanskrit documents== http://sanskritdocuments.org [[Category:Hinduism]] [[Category:Theology]] [[Category:Religion]] i8ty6eliw9smz2zgdfi1ozz9rvdglvx 0 53312 2806726 2803665 2026-04-26T22:44:38Z ~2026-25598-21 3068809 2806726 wikitext text/x-wiki [[File:Alice par John Tenniel 30.png|thumb|right|An illustration from Lewis Carroll's ''Alice's Adventures in Wonderland'', depicting the fictional protagonist, Alice, playing a fantastical game of croquet.]] [[File:AmericasBestComics2901.jpg|thumb]] [[File:Tarzan All Story.jpg|thumb]] [[File:Little Nemo Clowns2.jpg|thumb]] [[File:FairbanksMarkofZorro.jpg|thumb]] Fiction generally is a narrative form, in any medium, consisting of imaginary people, events, or places—in other words, not based strictly on history or fact. It also commonly refers, more narrowly, to written narratives in prose and often specifically novels. In film, it generally corresponds to narrative film in opposition to documentary.<ref>[[Wikipedia: Fiction]]</ref> == Resources == * [[Exploring science through fiction]] * [[Fiction writing support group]] * [[Science Fiction Challenge]] * [[Portal:Literary Studies]] * [[Fiction writing]] == List of fictions == * Cartoons (Mickey Mouse, Winnie the Pooh, Popeye, Betty Boop, Felix the Cat, Looney Tunes and Merrie Melodies, Woody Woodpecker, Rocky and Bullwinkle, The Flintstones, The Jetsons, Scooby-Doo, Mr. Magoo, Mighty Mouse, The Pink Panther, SpongeBob SquarePants, Rugrats, The Loud House, Oggy and the Cockroaches, The Simpsons, Futurama, South Park, Beavis and Butt-Head, King of the Hill, Rick and Morty, The Powerpuff Girls, Ben 10, etc.) * Video Games (Mario, Donkey Kong, Sonic the Hedgehog, PaRappa the Rapper, Pac-Man, Mega Man, The Legend of Zelda, Kirby, Crash Bandicoot, Rayman, Banjo Kazooie, Cuphead, Enchanted Portals, Angry Birds, Shantae, etc.) * Comics (Superman, Batman, Wonder Woman, Captain Marvel / Shazam, Spider-Man, The Incredible Hulk, Garfield, The Smurfs, Peanuts, Blondie, Spirou, Marsupilami, The Katzenjammer Kids, Little Nemo in Slumberland, Yakari, Tintin, Heathcliff, Cubitus, Dilbert, Teenage Mutant Ninja Turtles, Rupert Bear, Dennis the Menace (Hank Ketcham), Dennis the Menace and Gnasher, Archie, Baby Blues, Asterix, Lucky Luke, The Phantom, Buck Rogers, Flash Gordon, etc.) * Anime / Manga (Dragon Ball, Pokémon, Doraemon, Astro Boy, Princess Knight, Maya the Bee, Vicky the Viking, Yu-Gi-Oh!, Sazae-san, Sailor Moon, Beyblade, Robotan, My Neighbor Totoro, FLCL, Tenkai Knights, Attack on Titan, Digimon, Naruto, Bleach, One Piece, Anpanman, etc.) * Films (Star Wars, Indiana Jones, Harry Potter, Underworld, Terminator, Jurassic Park, Kill Bill, The Godfather, Back to the Future, The Chronicles of Narnia, Planet of the Apes, The Lord of the Rings, The Hobbit, Pirates of the Caribbean, Mary Poppins, King Kong, Godzilla, Charlie Chaplin, Laurel and Hardy, The Marx Brothers, Rocky, Mad Max, Fast & Furious, James Bond, Nosferatu, Puppet Master, Mission: Impossible, Gone with the Wind, Police Academy, Jaws, The Exorcist, Saw, A Nightmare on Elm Street, Friday the 13th, Cleopatra, Buster Keaton, Ben-Hur, Child's Play, The Shining, Pulp Fiction, Full Metal Jacket, Men in Black, Don Juan, The Jazz Singer, Lights of New York, The Birth of a Nation, Ted, Titanic, Avatar, Casablanca, Rambo, The Matrix, Halloween, Hellraiser, Home Alone, Marilyn Monroe, Austin Powers, The Wizard of Oz, The Three Stooges, etc.) * Animated Films (Snow White and the Seven Dwarfs, Cinderella, The Little Mermaid, Aladdin, Pinocchio, Hercules, Lilo & Stitch, The Lion King, Gulliver's Travels, Anastasia, etc.) * Computer-Animations (Toy Story, Cars, Luxo Jr., Knick Knack, Tin Toy, Shrek, Ice Age, Frozen, Tangled, Moana, Despicable Me, Miraculous: Tales of Ladybug & Cat Noir, VeggieTales, Antz, A Bug's Life, Wonder Park, Monsters, Inc., Finding Nemo, The Incredibles, etc.) * Stop-Motion Animations (Wallace and Gromit, Shaun the Sheep, Coraline, Pingu, Roary the Racing Car, Bertha, Rudolph the Red-Nosed Reindeer, The Nightmare Before Christmas, Postman Pat, Fireman Sam, Bob the Builder, The Wombles, Gumby, The PJs, Davey and Goliath, etc.) * Literature (Frankenstein, Dracula, The Phantom of the Opera, Sherlock Holmes, Thomas & Friends, Babar the Elephant, Peter Rabbit, Paddington Bear, Where's Wally?, Raggedy Ann and Andy, Dr. Jekyll and Mr. Hyde, Around the World in Eighty Days, The War of the Worlds, Don Quixote, Conan the Barbarian, Nancy Drew, Madeline, The Cat in the Hat, Grinch, Noddy, Tarzan, Mr. Men and Little Miss, Richard Scarry's Busytown, Pride and Prejudice, Alice in Wonderland, Hercule Poirot, Miss Marple, Zorro, 20,000 Leagues Under the Sea, The Three Musketeers, Wuthering Heights, etc.) * Advertising (Ronald McDonald, M&M's, Tony the Tiger, Tetley Tea Folk, Pillsbury Doughboy, Kool-Aid Man, Noid, California Raisins, Jack in the Box, Green Giant, Sonny the Cuckoo Bird, Cap'n Crunch, etc.) * Greeting Cards (Strawberry Shortcake, Care Bears, Rainbow Brite, etc.) * Toys (Transformers, My Little Pony, Barbie, Masters of the Universe, G.I. Joe, etc.) * Radio Series (The Green Hornet, The Lone Ranger, The Archers, etc.) * Television Series (Star Trek, Doctor Who, Mr. Bean, Blackadder, Tugs, Mind Your Language, Monty Python, Between the Lions, The X-Files, Seinfeld, Game of Thrones, Sesame Street, The Muppets, Sam & Cat, The Bill, Benny Hill, Cops, Jackass, Happy Days, The Munsters, The Addams Family, As Time Goes By, Family Ties, Fawlty Towers, ALF, The Dukes of Hazzard, etc.) == See Also == * [[Wikipedia: Fiction]] * [[Wikibooks: Writing Adolescent Fiction]] == References == {{Reflist}} {{subpagesif}} [[Category:Reading]] kwe0urmlaq5igrya0nxmcs6ggy3bku2 4 75745 2806744 2806671 2026-04-27T04:10:02Z Dronebogus 3054149 /* AI-generated images */ Reply 2806744 wikitext text/x-wiki {{/Header}} == Dan Polansky == I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC) : The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC) :[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:03, 27 November 2025 (UTC) ::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC) == Sidewide count.js == i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.<br><br> so a "count.js" would complete it. See [[User:Harold Foppele/common.js]]. If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC) == need to add my profile == im trying to add new profile content [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC) :You can edit it now. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:05, 4 February 2026 (UTC) ::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) :::i have created and its in sandbox. i would like to know when it will be approved [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC) ::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC) == Im trying to add new profile while add content its shows not alowed == This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC) == New User: cannot create talk page == Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is: <blockquote> I have doubts as to to the reliability of this essay. Take for rexample the sentence: <blockquote> Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report<ref>Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., & Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next</ref>. </blockquote> Reading the reference (freely available on ResearchGate), one notes: # that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and # that the reference factually contradicts this essay. Quoting from the reference: <blockquote> There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year. </blockquote> Not that the reference says ''could save'', not ''saves'' as in the essay. This calls into question the reliability of the essay. </blockquote> Could an administrator make this addition for me? Thank you! {{reflist}} [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC) :@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC) ::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC) == Disallowed to add a page on a course == I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with "New User Exceeded New Page Limit". Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC) :Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC) == Harold Foppele adding LLM-generated nonsense and personal fiction == I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes: *[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise "nonfiction" article *Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM) *Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with. Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC) :Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright. :I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC) == Repeated removal of RFD notices by Harold Foppele == {{User|Harold Foppele }} This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already. They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that. I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 23:03, 4 March 2026 (UTC) : {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC) == Blocks for sockpuppet == Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC) :<small>@[[User:MathXplore|MathXplore]]</small> [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC) :: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC) :CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC) ::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&zwj;🇺🇦&nbsp;[[User:Timtrent|Timtrent]]&nbsp;🇺🇦&nbsp;[[User talk:Timtrent|talk to me]]&nbsp;🇺🇦&zwj;🇵🇸 11:41, 8 March 2026 (UTC) == Problem when trying to start a discussion with authors of the Plurilingual education portal == The authors I wanted to discuss with are called "Project PEP" and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC) :@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC) == Please publish my post == My post is about "Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence. With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care." [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC) :@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC) ::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC) :I looked at your attempts to add this text and I see a link to one website repeated many times, which reminds me of the misuse of Wikiversity for self-promotion or to increase the importance of the website. It is necessary to remind you here that Wikiversity is not a place for promotion, but a place for education. So if you want to educate, it will not be a problem to create the page without external links with a clearly defined procedure for how people should use it and what to expect from it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:07, 1 April 2026 (UTC) == New user limit == Hi, I am creating an AIPA Method learning resource page. I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&veaction=edit] Thank you for your help. Best regards, Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC) :@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is "new user limit", but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC) ::Thank you, you are very kind. ::I will wait a day, and try again (without links, too). ::Today, I already created About Me info page, and maybe that is enough for the filters for one day. [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC) :::Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:56, 1 April 2026 (UTC) ::::Hi, my About Me page is just an info page with the neutral as possible presentation of my work. ::::There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that. ::::There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: "I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University." I think that if you had written a book on Life Science, you would have mentioned that as well. ::::Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies. ::::Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users): [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:47, 2 April 2026 (UTC) :::::For the unknown reasons, the form didn't publish my second part of the message: :::::I believe this is a valid contribution to Wikiversity. :::::Best Regards, :::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:52, 2 April 2026 (UTC) ::::::And the third try: :::::: == Introduction == ::::::The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management. ::::::The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome. :::::: == Theoretical foundations == ::::::The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on: ::::::* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma & Metzinger’s large‑scale study of content‑reduced awareness states. ::::::* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness. ::::::* Narrative identity and partial‑self models within personality and identity theory. ::::::Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience. :::::: == Experiential empiricism == ::::::The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining: ::::::* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes. ::::::* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006). ::::::* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation. ::::::* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions. ::::::All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement. ::::::I believe this is a valid contribution to Wikiversity. ::::::Best regards, ::::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:54, 2 April 2026 (UTC) == Unable to publish pages == Whenever I try to publish a page with linked sources it gets flagged and says I'm a new user attempting to publish content with outside links. Those outside links are my sources. [[User:Soboyed|Soboyed]] ([[User talk:Soboyed|discuss]] • [[Special:Contributions/Soboyed|contribs]]) 04:52, 2 April 2026 (UTC) :This restriction is automatically lifted after you have edited for a certain time (I don't recall that time off-hand, but it is not long). This is designed to stop spam. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:53, 2 April 2026 (UTC) == Showing error to publish a Post == My action was constructive, not destructive, please allow to publish it. [[Special:Contributions/&#126;2026-20906-18|&#126;2026-20906-18]] ([[User talk:&#126;2026-20906-18|talk]]) 08:06, 4 April 2026 (UTC) :Maybe you got caught in a filter. Consider [[Special:CreateAccount|creating an account]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:06, 4 April 2026 (UTC) :Your edits, [https://en.wikiversity.org/w/index.php?title=Special:AbuseLog&wpSearchUser=%7E2026-20906-18 these ones], seems to have tripped a filter when you tried to create a page on [[Create]] which external links. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:58, 4 April 2026 (UTC) :Have you read my [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&diff=prev&oldid=2802219 previous reply] to you @[[User:~2026-20906-18|~2026-20906-18]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:02, 6 April 2026 (UTC) == Abuse filters which should be deleted == Hi, there are some abuse filters which should probably be deleted. * [[Special:AbuseFilter/1]] (not needed anymore) * [[Special:AbuseFilter/2]] (no hits since 2018) * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) * [[Special:AbuseFilter/5]] (no hits since 2023) * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) * [[Special:AbuseFilter/17]] (no hits since 2022) * [[Special:AbuseFilter/19]] (no hits since 2019) * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:51, 5 April 2026 (UTC) :Why do these need to be deleted rather than inactivated? Do inactive abuse filters cause a server strain? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:39, 5 April 2026 (UTC) :: Deleted filters do not cause strain to the servers. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:28, 5 April 2026 (UTC) :These sounds like sensible suggestions but, yes, would inactivation perhaps make more sense than deletion for at least some filters? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:35, 5 April 2026 (UTC) :I would keep them @[[User:Codename Noreste|Codename Noreste]]. Alternatively, I would turn off the ones that haven't caught anything for a long time, but I would leave them enabled in case they need to be turned on or improved. If someone has already written the code and we don't have hundreds of free man-hours of programmers on Wikiversity, the server load seems secondary to me, and is negligible compared to other things. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:11, 6 April 2026 (UTC) :: I know how to write abuse filter code and regex, but I would recommend disabling filters that have never caught anything in a long time ''and'' those who made lots of false positives. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:09, 6 April 2026 (UTC) :::Of course @[[User:Codename Noreste|Codename Noreste]], there are people here today who are capable of changing the code. But the question is what it will be like in a few years, the question is what will happen if those two are busy for a long time, etc. That's why I would leave it so that those who don't know much about code can be inspired by it and will need to do something with it someday - plus, more code for different types of filtering is actually great educational material on how those filters work. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:41, 17 April 2026 (UTC) Here's the updated list of abuse filters under review with actions I've taken (several disabled, one basic code improvement, and some actions changed) - none have been deleted so they can all be edited and reactivayed - please suggest any further changes: * [[Special:AbuseFilter/1]] (not needed anymore) - One time account spam bot - 4 hits over 10 years ago - Disabled in 2024 - May be useful in future * [[Special:AbuseFilter/2]] (no hits since 2018) - Userspace spamming - 778 hits; none since 2018 likely due to global filters - Now disabled * [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) - Specific user page spam - 1,101 hits most recent 7 March 2026 - Still active - Kept enabled * [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) - Questionable Language (profanity) - 6,055 hits including very recently - However it was logging hits without taking any actions - Edited to reduce likelihood of false positives by only applying filter to users with low (< 20) edit count and applied weak actions to tag and warn but not prevent publishing the content * [[Special:AbuseFilter/5]] (no hits since 2023) - Blocked Solicitation Links - 95 hits; none since 2023 - blocks specific historical spam sites - Non-active - Now disabled * Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) - Not reviewed - They are currently disabled * [[Special:AbuseFilter/17]] (no hits since 2022) - Fundamental Physics Edits - 347 hits; none since 2022 - Non-active and very specific for a historical issue - Now disabled * [[Special:AbuseFilter/19]] (no hits since 2019) - Page Creation - 20 hits; none since 2019 - Retained for historical reference and possible future updates - Now disabled * [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) - Globally Banned Editor (renamed to Low-edit Spam Monitor) - 2,829 hits including very recent - Only applies to users with less than 5 edits and takes no actions / monitoring only - Reviewing the details of the hits I don't see many false positives and have strengthened its actions to add a tag and warning -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:57, 9 April 2026 (UTC) == Block request == Please block ~2026-20985-80/~2026-21079-90/~2026-21223-88. Reason: Vandalism. [[User:Àncilu|Àncilu]] ([[User talk:Àncilu|discuss]] • [[Special:Contributions/Àncilu|contribs]]) 23:24, 5 April 2026 (UTC) :All edits should be deleted and the first is blocked by Atcovi. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:33, 6 April 2026 (UTC) == Antispam - Filter 12 == {{ping|Codename Noreste}} Thanks for contacting me with a suggested [[Special:AbuseFilter|abuse filter]] for the coupon spam we've been getting. A very much appreciated time saver. Per your suggestion, abuse filter 12 has been reactivated with your updated regex. It should tag and prevent page creation actions for coupon promo etc. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:33, 15 April 2026 (UTC) == Urgent! error message This action has been automatically identified as potentially harmful, and therefore disallowed == While submitting the post this error was coming "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link" How to resolve it? Here is the content: {{note|marketing material removed}} [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:14, 17 April 2026 (UTC) : That's because Wikiversity doesn't accept advertising. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:20, 17 April 2026 (UTC) ::So this is fine ::'''Online Internship and Digital Learning for Students''' ::Online learning has become an important part of modern education. With the help of the internet and digital tools, students can now study, practice, and gain experience without being physically present in a classroom. One key part of this system is the online internship, which helps students learn real-world skills along with their studies. ::An online internship allows students to work on tasks and projects through digital platforms. This makes learning more practical and useful, especially for those who want to understand how real work environments function. ::'''Background''' ::The concept of online learning developed from distance education, where students learned from remote locations. Over time, with the growth of digital technology, learning has become more interactive and flexible. ::The introduction of the online internship has added another important layer to digital education. It combines theoretical learning with practical experience, helping students prepare for future careers. ::During global events such as the COVID-19 pandemic, online education and online internship programs became essential. They helped students continue learning and gaining experience despite restrictions on physical movement. ::'''Importance of Online Internship''' ::An online internship plays an important role in student development. It helps bridge the gap between academic knowledge and practical skills. ::'''Some key points include:''' ::Students understand how real work is done ::They develop basic professional skills ::It supports career readiness ::It allows learning without location limits ::By participating in an online internship, students can improve their confidence and gain early exposure to different fields. ::'''Features of Online Internship-Based Learning''' ::Modern education platforms often include online internship opportunities as part of their learning system. These usually offer: ::Flexible schedules for students ::Access to learning from home ::Beginner-friendly tasks and projects ::A combination of theory and practice ::Such features make online internship programs suitable for a wide range of learners, including beginners. ::'''Learning Tasks''' ::Explore how online internship programs support student learning in digital environments. ::Identify how students can gain practical experience through an online internship ::Analyze the role of flexible learning in improving student engagement ::Understand how online education and internships work together ::'''References''' ::[https://ijpsl.in/wp-content/uploads/2020/09/E-Learning_Hanaaya-Navaneeth.pdf The Past, Present and Future of E-Learning: Hanaaya Varyani and Navaneeth M S] ::[https://ieeexplore.ieee.org/document/9788102 Current Trends and Future Perspectives of e-Learning in India] ::'''See Also''' ::[https://easyshiksha.com/online_courses/internship Online Internship] ::[https://easyshiksha.com/online_courses/ Online Courses] ::[https://easyshiksha.com/online_courses/kids-learning Kids Learning] ::[https://easyshiksha.com/career_helper/ Career Guidance] [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:27, 17 April 2026 (UTC) :::@[[User:EasyshikshaMarketing|EasyshikshaMarketing]] Wikiversity is a resource for education, or a space for education. However, your intention to link to another website is obvious, and such content does not belong here, as it contradicts the purpose of Wikiversity. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:38, 17 April 2026 (UTC) ==AI-generated images== Seeking your advice. Myself and students use some AI-generated images in the [[Motivation and emotion]] project. [[User:Dronebogus]] has been removing some of these images from Wikiversity pages and nominating them for deletion at Commons. As a result, some have been deleted and some have been kept. Dronebogus has made some useful edits and image suggestions for [[Motivation and emotion]] which I've appreciated and incorporated. However, there are other edits to remove an AI image by Dronebogus that I've reverted where I think the image is more educational than no image or an alternative image suggested by Dronebogus. There are a couple of pages where Dronebogus has reverted my reversion, so we are at risk of edit warring. We have briefly discussed and warned each other on our user talk pages, but it seems to come down to a difference in perception about the educational usefulness of the AI images. So, I'm asking here for others to please review the recent edit histories for these pages: * [[Motivation and emotion/Lectures/Brain and physiological needs]] * [[Motivation and emotion/Book/2025/Stockholm syndrome emotion]] and let us know what you think about the AI image suitability vs. using no image or alternative images suggested by Dronebogus. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 26 April 2026 (UTC) :I think Jtneill needs to try harder to find non-AI alternatives both on Commons and the web. I’m not reiterating the well known problems with generative AI— you can read about those on Wikipedia and the broader Internet. Needless to say it’s kind of just inherently toxic. If you use it, it should be the last resort of last resorts. Just my stance, which I consider perfectly valid and reasonable. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:10, 27 April 2026 (UTC) tsikrvbjn66gvqw9bdal3ir32xj0h2s 0 98602 2806683 2806650 2026-04-26T16:49:33Z Juandev 2651 Undid revision [[Special:Diff/2806650|2806650]] by [[Special:Contributions/Dronebogus|Dronebogus]] ([[User talk:Dronebogus|talk]]) 2806683 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 03: Brain and physiological needs|third}} {{Motivation and emotion/Lectures/Complete}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:WP20Symbols brain.svg|250px|right]] ==Overview== This lecture: * explains the role of [[Motivation and emotion/Brain structures|brain structures]], [[Motivation and emotion/Neurotransmitters|neurotransmitters]], and [[Motivation and emotion/Hormones|hormones]] in regulating motivational drives * discusses physiological needs, particularly thirst, hunger, and sexual motivation Take-home messages: * The brain is as much about motivation and emotion as it is about cognition and thinking * Biological urges are underestimated motivational forces when we are not currently experiencing them ==Outline== [[File:Man with superimposed brain.jpg|thumb|What is the brain's involvement in [[motivation and emotion]]? It's easy to ignore the brain's role in motivation and emotion in part because its covered by bone, skin, hair, and adornments. What if our brains were more observable?]] [[File:Hunger strike - Day 53.JPG|thumb|right|290px|Physiological needs<!-- such as breathing, drinking, urinating, eating, defecating, and sleeping--> are often overlooked as motivational forces until they fall outside of [[w:Homeostasis|homeostasis]] and become increasingly urgent. It takes extreme motivation, for example, to go on an extended hunger strike.]] ;Motivated and emotional brain * Neuroscience * Brain structures * Subcortical ** Reticular formation ** Amygdala **Reward centre **Basal ganglia **Hypothalamus * Cortical ** Insula ** Prefrontal cortex ** Orbitofrontal cortex ** Ventromedial PFC ** Dorsolateral PFC ** Anterior cingulate cortex * Bidirectional ** Neurotransmitters ** Dopamine ** Serotonin ** Norepinephrine ** Endorphins *Hormones ** Cortisol ** Oxytocin ** Testosterone ** Ghrelin (Part B) ** Leptin (Part B) ;Physiological needs * Needs * Regulatory processes * Example physiological needs ** Thirst ** Hunger ** Sexual motivation ==Focus== This lecture highlights specific brain structures and communication pathways that psychological science has identified as contributing to the subjective experience of various motivational and emotional states. ==3D brain model== * Learn about the location and function of key brain structures using [https://www.brainfacts.org/3d-brain 3d brain] (brainfacts.org) * This 3D, interactive model of the human brain shows the main structures and explains their functions. * Task: Can you find each of the brain structures mentioned in this lecture in the 3D model? ==Readings== * Chapter 03: The motivated and emotional brain ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]] or [[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) * Chapter 04: Physiological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 4: Biological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2024]]) ==Slides== <!-- ** [https://docs.google.com/presentation/d/1wNaegpzIkQ4XyeRcN9BRXQ1gGNR5XX3cG7x_dtBGj6c/edit?usp=sharing Lecture 01 and 02 recap] (Google Slides) --> * [https://docs.google.com/presentation/d/1oI8g-0xvSxETUwYOW1TLsRJdiSq3AbVq6YMlm8D3ivc/edit?usp=sharing Motivated and emotional brain] (Google Slides) * [https://docs.google.com/presentation/d/1LgYQ9ydIaj5AJZEW7MkH1M2zVKxjWQe4vetZnOairQE/edit?usp=sharing Physiological needs] (Google Slides) <!-- ** [https://www.slideshare.net/jtneill/motivation-and-emotion-introduction-and-historical-perspectives-recap Lecture 01 and 02 recap] (Slideshare) ** [https://www.slideshare.net/jtneill/motivated-and-emotional-brain Motivated and emotional brain] (Slideshare) ** [https://www.slideshare.net/jtneill/physiological-needs Physiological needs] (Slideshare) --> <!-- * [http://www.slideshare.net/jtneill/brain-and-physiological-needs Lecture slides] (Slideshare) * Handouts ** [[Media:Brain and physiological needs 6 slides per page.pdf|Download 6 slides to a page]]: [[File:Brain and physiological needs 6 slides per page.pdf|100px]] ** [[Media:Brain and physiological needs 3 slides per page.pdf|Download 3 slides to a page]]:[[File:Brain and physiological needs 3 slides per page.pdf|100px]] --> ==See also== ;Wikiversity * [[/Images/]] * [[Motivation and emotion/Brain structures|Brain structures]] * [[Motivation and emotion/Hormones|Hormones]] * [[Motivation and emotion/Neurotransmitters|Neurotransmitters]] * Book chapters ** [[:Category:Motivation and emotion/Book/Brain|Brain]] (Category) ** [[:Category:Motivation and emotion/Book/Hormones|Hormones]] (Category) ** [[:Category:Motivation and emotion/Book/Neurotransmitters|Neurotransmitters]] (Category) ** [[:Category:Motivation and emotion/Book/Needs/Physiological|Physiological needs]] (Category)<!-- [[Motivation and emotion/Book/2025/Thirst regulation|Thirst regulation]] --> ;Wikipedia * [[w:Autonomic nervous system|Autonomic nervous system]] * [[w:ERG theory|ERG theory]] * [[w:Limbic system|Limbic system]] * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] * [[w:Nucleus (neuroanatomy)|Nucleus (neuroanatomy)]] * [[w:Parasympathetic nervous system|Parasympathetic nervous system]] * [[w:Prefrontal cortex|Prefrontal cortex]] * [[w:Reward system|Reward system]] * [[w:Sympathetic nervous system|Sympathetic nervous system]] ;Lectures * [[{{#titleparts:{{PAGENAME}}|2}}/Historical development and assessment skills|Historical development and assessment skills]] (Previous lecture) * [[{{#titleparts:{{PAGENAME}}|2}}/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] (Next lecture) ;Tutorials * [[Motivation and emotion/Tutorials/Physiological needs|Physiological needs]] <!-- ==References== {{Hanging indent|1= Australian Bureau of Statistics (2013). [http://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/4338.0~2011-13~Main%20Features~Overweight%20and%20obesity~10007 Overweight and obesity]. ''4338.0 - Profiles of Health, Australia, 2011-13''. Eder, A. B., Elliot, A. J., & Harmon-Jones, E. (2013). [http://emr.sagepub.com/content/5/3/227 Approach and avoidance motivation: Issues and advances]. ''Emotion Review'', ''5''(3), 308-311. https://doi.org/10.1177/1754073913477990.}} --> ==Recording== * [https://au-lti.bbcollab.com/recording/54f3cdb5b30a476fbcbb77824a1b9dfb Lecture 03] (2025)<!-- * [https://au-lti.bbcollab.com/recording/b8834e9830314aa3b804d3c6c3e7a740 Lecture 03] (2024) * [https://au-lti.bbcollab.com/recording/546476bf547f4efd8ae55b05e4547efc Lecture 03] (2023) * [https://au-lti.bbcollab.com/recording/17f200f050e044da9a6571ffdf63c78c Lecture 03] (2022) * [https://au-lti.bbcollab.com/recording/d34da988d75c48b99df662329594cc9f Lecture 03] (2021) --> ==References== {{Hanging indent|1= Saper, C. B., & Lowell, B. B. (2014). The hypothalamus. ''Current Biology'', ''24''(23), R1111–R1116. https://doi.org/10.1016/j.cub.2014.10.023 }} ==External links== * [https://fs.blog/knowledge-project-podcast/anna-lembke/ Between pleasure and pain] (Dr. Anna Lembke, The Knowledge Project Ep. #159) * [https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/short-stuff-hangry-102038598/ Hangry] (Stuff You Should Know, Podcast, 12:30 mins) * [https://www.youtube.com/watch?v=tZ4YnYUJnOQ&list=PL9JAHwJN4qyArhEyLUgU_MoGddk2PVTeb Hormones of hunger: Leptin and ghrelin] (Corporis, 2019, YouTube, 9:33 mins) - how leptin and ghrelin work together to modulate hunger<!-- As you watch the video, consider: What causes hunger and eating? --> * [https://www.ted.com/playlists/1/how_does_my_brain_work How does my brain work?] (TED Talks playlist) * [https://www.youtube.com/watch?v=Qymp_VaFo9M Let's talk about sex] (Crash Course Psychology #27; YouTube 11:35 mins) * [https://www.ted.com/talks/david_anderson_your_brain_is_more_than_a_bag_of_chemicals Your brain is more than a bag of chemicals] (David Anderson, 2013, TED talk, 16 mins) - neuroscientific research into motivation and emotion using a basic animal model (fruit fly)<!-- As you watch the video, some questions to think about: 1. Do animals experience emotions? If so, which emotions - and why? 2. What might pharmacological treatment of psychological disorders look like in 20, 50, 100 years? --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Brain and physiological needs]] rnq5boy2iwmsiyglk1cdyqtks1ork07 2806747 2806683 2026-04-27T04:11:11Z Dronebogus 3054149 Undid revision [[Special:Diff/2806683|2806683]] by [[Special:Contributions/Juandev|Juandev]] ([[User talk:Juandev|talk]]) this image isn’t educational! At least replace it with a less god-awful decorative image 2806747 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 03: Brain and physiological needs|third}} {{Motivation and emotion/Lectures/Complete}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:WP20Symbols brain.svg|250px|right]] ==Overview== This lecture: * explains the role of [[Motivation and emotion/Brain structures|brain structures]], [[Motivation and emotion/Neurotransmitters|neurotransmitters]], and [[Motivation and emotion/Hormones|hormones]] in regulating motivational drives * discusses physiological needs, particularly thirst, hunger, and sexual motivation Take-home messages: * The brain is as much about motivation and emotion as it is about cognition and thinking * Biological urges are underestimated motivational forces when we are not currently experiencing them ==Outline== [[File:Brain icon from Noun Project.png |thumb|What is the brain's involvement in [[motivation and emotion]]? It seems easy to "ignore" the brain's role in psychological experience in part because its visually hidden under the skull which is covered by skin, hair, and adornments. But what if our brains were more observable, on the outside?]] [[File:Hunger strike - Day 53.JPG|thumb|right|290px|Physiological needs such as breathing, drinking, urinating, eating, defecating, and sleeping are often overlooked as motivational forces until they range outside of [[w:Homeostasis|homeostasis]] and then become increasingly urgemt amd motivationally demanding. It takes extreme motivation, for example, to go on an extended hunger strike.]] ;Motivated and emotional brain * Neuroscience * Brain structures * Subcortical ** Reticular formation ** Amygdala **Reward centre **Basal ganglia **Hypothalamus * Cortical ** Insula ** Prefrontal cortex ** Orbitofrontal cortex ** Ventromedial PFC ** Dorsolateral PFC ** Anterior cingulate cortex * Bidirectional ** Neurotransmitters ** Dopamine ** Serotonin ** Norepinephrine ** Endorphins *Hormones ** Cortisol ** Oxytocin ** Testosterone ** Ghrelin (Part B) ** Leptin (Part B) ;Physiological needs * Needs * Regulatory processes * Example physiological needs ** Thirst ** Hunger ** Sexual motivation ==Focus== This lecture highlights specific brain structures and communication pathways that psychological science has identified as contributing to the subjective experience of various motivational and emotional states. ==3D brain model== * Learn about the location and function of key brain structures using [https://www.brainfacts.org/3d-brain 3d brain] (brainfacts.org) * This 3D, interactive model of the human brain shows the main structures and explains their functions. * Task: Can you find each of the brain structures mentioned in this lecture in the 3D model? ==Readings== * Chapter 03: The motivated and emotional brain ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]] or [[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) * Chapter 04: Physiological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 4: Biological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2024]]) ==Slides== <!-- ** [https://docs.google.com/presentation/d/1wNaegpzIkQ4XyeRcN9BRXQ1gGNR5XX3cG7x_dtBGj6c/edit?usp=sharing Lecture 01 and 02 recap] (Google Slides) --> * [https://docs.google.com/presentation/d/1oI8g-0xvSxETUwYOW1TLsRJdiSq3AbVq6YMlm8D3ivc/edit?usp=sharing Motivated and emotional brain] (Google Slides) * [https://docs.google.com/presentation/d/1LgYQ9ydIaj5AJZEW7MkH1M2zVKxjWQe4vetZnOairQE/edit?usp=sharing Physiological needs] (Google Slides) <!-- ** [https://www.slideshare.net/jtneill/motivation-and-emotion-introduction-and-historical-perspectives-recap Lecture 01 and 02 recap] (Slideshare) ** [https://www.slideshare.net/jtneill/motivated-and-emotional-brain Motivated and emotional brain] (Slideshare) ** [https://www.slideshare.net/jtneill/physiological-needs Physiological needs] (Slideshare) --> <!-- * [http://www.slideshare.net/jtneill/brain-and-physiological-needs Lecture slides] (Slideshare) * Handouts ** [[Media:Brain and physiological needs 6 slides per page.pdf|Download 6 slides to a page]]: [[File:Brain and physiological needs 6 slides per page.pdf|100px]] ** [[Media:Brain and physiological needs 3 slides per page.pdf|Download 3 slides to a page]]:[[File:Brain and physiological needs 3 slides per page.pdf|100px]] --> ==See also== ;Wikiversity * [[/Images/]] * [[Motivation and emotion/Brain structures|Brain structures]] * [[Motivation and emotion/Hormones|Hormones]] * [[Motivation and emotion/Neurotransmitters|Neurotransmitters]] * Book chapters ** [[:Category:Motivation and emotion/Book/Brain|Brain]] (Category) ** [[:Category:Motivation and emotion/Book/Hormones|Hormones]] (Category) ** [[:Category:Motivation and emotion/Book/Neurotransmitters|Neurotransmitters]] (Category) ** [[:Category:Motivation and emotion/Book/Needs/Physiological|Physiological needs]] (Category)<!-- [[Motivation and emotion/Book/2025/Thirst regulation|Thirst regulation]] --> ;Wikipedia * [[w:Autonomic nervous system|Autonomic nervous system]] * [[w:ERG theory|ERG theory]] * [[w:Limbic system|Limbic system]] * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] * [[w:Nucleus (neuroanatomy)|Nucleus (neuroanatomy)]] * [[w:Parasympathetic nervous system|Parasympathetic nervous system]] * [[w:Prefrontal cortex|Prefrontal cortex]] * [[w:Reward system|Reward system]] * [[w:Sympathetic nervous system|Sympathetic nervous system]] ;Lectures * [[{{#titleparts:{{PAGENAME}}|2}}/Historical development and assessment skills|Historical development and assessment skills]] (Previous lecture) * [[{{#titleparts:{{PAGENAME}}|2}}/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] (Next lecture) ;Tutorials * [[Motivation and emotion/Tutorials/Physiological needs|Physiological needs]] <!-- ==References== {{Hanging indent|1= Australian Bureau of Statistics (2013). [http://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/4338.0~2011-13~Main%20Features~Overweight%20and%20obesity~10007 Overweight and obesity]. ''4338.0 - Profiles of Health, Australia, 2011-13''. Eder, A. B., Elliot, A. J., & Harmon-Jones, E. (2013). [http://emr.sagepub.com/content/5/3/227 Approach and avoidance motivation: Issues and advances]. ''Emotion Review'', ''5''(3), 308-311. https://doi.org/10.1177/1754073913477990.}} --> ==Recording== * [https://au-lti.bbcollab.com/recording/54f3cdb5b30a476fbcbb77824a1b9dfb Lecture 03] (2025)<!-- * [https://au-lti.bbcollab.com/recording/b8834e9830314aa3b804d3c6c3e7a740 Lecture 03] (2024) * [https://au-lti.bbcollab.com/recording/546476bf547f4efd8ae55b05e4547efc Lecture 03] (2023) * [https://au-lti.bbcollab.com/recording/17f200f050e044da9a6571ffdf63c78c Lecture 03] (2022) * [https://au-lti.bbcollab.com/recording/d34da988d75c48b99df662329594cc9f Lecture 03] (2021) --> ==References== {{Hanging indent|1= Saper, C. B., & Lowell, B. B. (2014). The hypothalamus. ''Current Biology'', ''24''(23), R1111–R1116. https://doi.org/10.1016/j.cub.2014.10.023 }} ==External links== * [https://fs.blog/knowledge-project-podcast/anna-lembke/ Between pleasure and pain] (Dr. Anna Lembke, The Knowledge Project Ep. #159) * [https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/short-stuff-hangry-102038598/ Hangry] (Stuff You Should Know, Podcast, 12:30 mins) * [https://www.youtube.com/watch?v=tZ4YnYUJnOQ&list=PL9JAHwJN4qyArhEyLUgU_MoGddk2PVTeb Hormones of hunger: Leptin and ghrelin] (Corporis, 2019, YouTube, 9:33 mins) - how leptin and ghrelin work together to modulate hunger<!-- As you watch the video, consider: What causes hunger and eating? --> * [https://www.ted.com/playlists/1/how_does_my_brain_work How does my brain work?] (TED Talks playlist) * [https://www.youtube.com/watch?v=Qymp_VaFo9M Let's talk about sex] (Crash Course Psychology #27; YouTube 11:35 mins) * [https://www.ted.com/talks/david_anderson_your_brain_is_more_than_a_bag_of_chemicals Your brain is more than a bag of chemicals] (David Anderson, 2013, TED talk, 16 mins) - neuroscientific research into motivation and emotion using a basic animal model (fruit fly)<!-- As you watch the video, some questions to think about: 1. Do animals experience emotions? If so, which emotions - and why? 2. What might pharmacological treatment of psychological disorders look like in 20, 50, 100 years? --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Brain and physiological needs]] eer515lpv8ayiqagr7avqe2kt8n08dy 2806748 2806747 2026-04-27T04:12:01Z Dronebogus 3054149 /* Outline */ 2806748 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 03: Brain and physiological needs|third}} {{Motivation and emotion/Lectures/Complete}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:WP20Symbols brain.svg|250px|right]] ==Overview== This lecture: * explains the role of [[Motivation and emotion/Brain structures|brain structures]], [[Motivation and emotion/Neurotransmitters|neurotransmitters]], and [[Motivation and emotion/Hormones|hormones]] in regulating motivational drives * discusses physiological needs, particularly thirst, hunger, and sexual motivation Take-home messages: * The brain is as much about motivation and emotion as it is about cognition and thinking * Biological urges are underestimated motivational forces when we are not currently experiencing them ==Outline== [[File:Human head and brain diagram.svg |thumb|What is the brain's involvement in [[motivation and emotion]]? It seems easy to "ignore" the brain's role in psychological experience in part because its visually hidden under the skull which is covered by skin, hair, and adornments. But what if our brains were more observable, on the outside?]] [[File:Hunger strike - Day 53.JPG|thumb|right|290px|Physiological needs such as breathing, drinking, urinating, eating, defecating, and sleeping are often overlooked as motivational forces until they range outside of [[w:Homeostasis|homeostasis]] and then become increasingly urgemt amd motivationally demanding. It takes extreme motivation, for example, to go on an extended hunger strike.]] ;Motivated and emotional brain * Neuroscience * Brain structures * Subcortical ** Reticular formation ** Amygdala **Reward centre **Basal ganglia **Hypothalamus * Cortical ** Insula ** Prefrontal cortex ** Orbitofrontal cortex ** Ventromedial PFC ** Dorsolateral PFC ** Anterior cingulate cortex * Bidirectional ** Neurotransmitters ** Dopamine ** Serotonin ** Norepinephrine ** Endorphins *Hormones ** Cortisol ** Oxytocin ** Testosterone ** Ghrelin (Part B) ** Leptin (Part B) ;Physiological needs * Needs * Regulatory processes * Example physiological needs ** Thirst ** Hunger ** Sexual motivation ==Focus== This lecture highlights specific brain structures and communication pathways that psychological science has identified as contributing to the subjective experience of various motivational and emotional states. ==3D brain model== * Learn about the location and function of key brain structures using [https://www.brainfacts.org/3d-brain 3d brain] (brainfacts.org) * This 3D, interactive model of the human brain shows the main structures and explains their functions. * Task: Can you find each of the brain structures mentioned in this lecture in the 3D model? ==Readings== * Chapter 03: The motivated and emotional brain ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]] or [[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) * Chapter 04: Physiological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 4: Biological needs ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2024]]) ==Slides== <!-- ** [https://docs.google.com/presentation/d/1wNaegpzIkQ4XyeRcN9BRXQ1gGNR5XX3cG7x_dtBGj6c/edit?usp=sharing Lecture 01 and 02 recap] (Google Slides) --> * [https://docs.google.com/presentation/d/1oI8g-0xvSxETUwYOW1TLsRJdiSq3AbVq6YMlm8D3ivc/edit?usp=sharing Motivated and emotional brain] (Google Slides) * [https://docs.google.com/presentation/d/1LgYQ9ydIaj5AJZEW7MkH1M2zVKxjWQe4vetZnOairQE/edit?usp=sharing Physiological needs] (Google Slides) <!-- ** [https://www.slideshare.net/jtneill/motivation-and-emotion-introduction-and-historical-perspectives-recap Lecture 01 and 02 recap] (Slideshare) ** [https://www.slideshare.net/jtneill/motivated-and-emotional-brain Motivated and emotional brain] (Slideshare) ** [https://www.slideshare.net/jtneill/physiological-needs Physiological needs] (Slideshare) --> <!-- * [http://www.slideshare.net/jtneill/brain-and-physiological-needs Lecture slides] (Slideshare) * Handouts ** [[Media:Brain and physiological needs 6 slides per page.pdf|Download 6 slides to a page]]: [[File:Brain and physiological needs 6 slides per page.pdf|100px]] ** [[Media:Brain and physiological needs 3 slides per page.pdf|Download 3 slides to a page]]:[[File:Brain and physiological needs 3 slides per page.pdf|100px]] --> ==See also== ;Wikiversity * [[/Images/]] * [[Motivation and emotion/Brain structures|Brain structures]] * [[Motivation and emotion/Hormones|Hormones]] * [[Motivation and emotion/Neurotransmitters|Neurotransmitters]] * Book chapters ** [[:Category:Motivation and emotion/Book/Brain|Brain]] (Category) ** [[:Category:Motivation and emotion/Book/Hormones|Hormones]] (Category) ** [[:Category:Motivation and emotion/Book/Neurotransmitters|Neurotransmitters]] (Category) ** [[:Category:Motivation and emotion/Book/Needs/Physiological|Physiological needs]] (Category)<!-- [[Motivation and emotion/Book/2025/Thirst regulation|Thirst regulation]] --> ;Wikipedia * [[w:Autonomic nervous system|Autonomic nervous system]] * [[w:ERG theory|ERG theory]] * [[w:Limbic system|Limbic system]] * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] * [[w:Nucleus (neuroanatomy)|Nucleus (neuroanatomy)]] * [[w:Parasympathetic nervous system|Parasympathetic nervous system]] * [[w:Prefrontal cortex|Prefrontal cortex]] * [[w:Reward system|Reward system]] * [[w:Sympathetic nervous system|Sympathetic nervous system]] ;Lectures * [[{{#titleparts:{{PAGENAME}}|2}}/Historical development and assessment skills|Historical development and assessment skills]] (Previous lecture) * [[{{#titleparts:{{PAGENAME}}|2}}/Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] (Next lecture) ;Tutorials * [[Motivation and emotion/Tutorials/Physiological needs|Physiological needs]] <!-- ==References== {{Hanging indent|1= Australian Bureau of Statistics (2013). [http://www.abs.gov.au/ausstats/abs@.nsf/Lookup/by%20Subject/4338.0~2011-13~Main%20Features~Overweight%20and%20obesity~10007 Overweight and obesity]. ''4338.0 - Profiles of Health, Australia, 2011-13''. Eder, A. B., Elliot, A. J., & Harmon-Jones, E. (2013). [http://emr.sagepub.com/content/5/3/227 Approach and avoidance motivation: Issues and advances]. ''Emotion Review'', ''5''(3), 308-311. https://doi.org/10.1177/1754073913477990.}} --> ==Recording== * [https://au-lti.bbcollab.com/recording/54f3cdb5b30a476fbcbb77824a1b9dfb Lecture 03] (2025)<!-- * [https://au-lti.bbcollab.com/recording/b8834e9830314aa3b804d3c6c3e7a740 Lecture 03] (2024) * [https://au-lti.bbcollab.com/recording/546476bf547f4efd8ae55b05e4547efc Lecture 03] (2023) * [https://au-lti.bbcollab.com/recording/17f200f050e044da9a6571ffdf63c78c Lecture 03] (2022) * [https://au-lti.bbcollab.com/recording/d34da988d75c48b99df662329594cc9f Lecture 03] (2021) --> ==References== {{Hanging indent|1= Saper, C. B., & Lowell, B. B. (2014). The hypothalamus. ''Current Biology'', ''24''(23), R1111–R1116. https://doi.org/10.1016/j.cub.2014.10.023 }} ==External links== * [https://fs.blog/knowledge-project-podcast/anna-lembke/ Between pleasure and pain] (Dr. Anna Lembke, The Knowledge Project Ep. #159) * [https://www.iheart.com/podcast/105-stuff-you-should-know-26940277/episode/short-stuff-hangry-102038598/ Hangry] (Stuff You Should Know, Podcast, 12:30 mins) * [https://www.youtube.com/watch?v=tZ4YnYUJnOQ&list=PL9JAHwJN4qyArhEyLUgU_MoGddk2PVTeb Hormones of hunger: Leptin and ghrelin] (Corporis, 2019, YouTube, 9:33 mins) - how leptin and ghrelin work together to modulate hunger<!-- As you watch the video, consider: What causes hunger and eating? --> * [https://www.ted.com/playlists/1/how_does_my_brain_work How does my brain work?] (TED Talks playlist) * [https://www.youtube.com/watch?v=Qymp_VaFo9M Let's talk about sex] (Crash Course Psychology #27; YouTube 11:35 mins) * [https://www.ted.com/talks/david_anderson_your_brain_is_more_than_a_bag_of_chemicals Your brain is more than a bag of chemicals] (David Anderson, 2013, TED talk, 16 mins) - neuroscientific research into motivation and emotion using a basic animal model (fruit fly)<!-- As you watch the video, some questions to think about: 1. Do animals experience emotions? If so, which emotions - and why? 2. What might pharmacological treatment of psychological disorders look like in 20, 50, 100 years? --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Brain and physiological needs]] m8umjcjgtw9ndr8shk7huoo9vip8482 0 132403 2806772 2793818 2026-04-27T09:46:08Z Д.Ильин 513564 2806772 wikitext text/x-wiki This is a '''Protein Crystallography Project''' concerned with the ''Determination of Protein Structures in Crystals and in Solutions with electrolytes by [[X-ray Diffraction]], [[neutron diffraction]], [[light scattering]], [[Vibrational Circular Dichroism]] (VCD) and 2D-NMR.'' ==Course on Protein Biochemistry== ==Course and Lecture Notes on X-ray Diffraction by Protein Crystals== [[image:Cytochrome C Oxidase 1OCC in Membrane 2.png|thumb|600px|center| Model of '''Cytochrome C Oxidase 1OCC''' in a Membrane]] ==Course and Lecture Notes on Neutron Diffraction by Protein Crystals== ==Course and Lecture Notes on 2D-NMR Determination of Protein Structure in Solution== ===Lecture Notes of NMR on Magnetically-oriented Protein Microcrystals in suspension=== [[image:GWM HahnEchoDecay.gif|thumb|600px|center|HahnEchoDecay: '''click on image to start the animation''']] ==Course on Computer Modeling of Protein Structure and Dynamics== ===Lecture Notes on Paracrystal Theory and Quasi-crystals=== *'''''The following figures and data have the following''''' [[Wikibooks:An Introduction to Molecular Biology/Function and structure of Proteins|'''Source: ''An Introduction to Molecular Biology/Function and structure of Proteins''''']] [[image:Zinc finger rendered.png|thumb|center|600px|Cartoon representation of the zinc-finger motif of proteins. The zinc ion (green) is coordinated by two histidine and two cysteine amino acid residues. Based on the [http://www.pdb.org X-ray structure of PDB]]] [[image:InsulinMonomer.jpg|thumb|center|600px|Created by Isaac Yonemoto. created with en:pymol, en:inkscape, and en:gimp from NMR structure 1ai0 in the en:pdb. Ref: Chang, X., Jorgensen, A.M., Bardrum, P., Led, J.J.]] [[image:InsulinHexamer.jpg|thumb|center|600px|Created by Isaac Yonemoto created with en:pymol, en:inkscape, and en:gimp from NMR structure 1ai0 in the en:pdb. Ref: Chang, X., Jorgensen, A.M., Bardrum, P., Led, J.J.]] ==Amino Acid Structures and Properties== The 20 amino acids encoded directly by the genetic code can be divided into several groups based on their properties. Important factors are charge, hydrophilicity or hydrophobicity, size and functional groups.Amino acids are usually classified by the properties of their side chain into four groups. The side chain can make an amino acid a weak acid or a weak base, and a hydrophile if the side chain is polar or a hydrophobe if it is nonpolar. [[Image:a-amino-acid.png|thumb|150px|An α-amino acid. The C_αH atom is omitted in the diagram.]] Protein amino acids are combined into a single [[polypeptide chain]] in a [[condensation reaction]]. This reaction is [[catalysis|catalysed]] by the [[ribosome]] in a process known as [[peptide biosynthesis|translation]]. {| class="wikitable" ! Essential ! Nonessential |- | [[Isoleucine]] | [[Alanine]] |- | [[Leucine]] | [[Asparagine]] |- | [[Lysine]] | [[Aspartic Acid]] |- | [[Methionine]] | [[Cysteine]]* |- | [[Phenylalanine]] | [[Glutamic Acid]] |- | [[Threonine]] | [[Glutamine]]* |- | [[Tryptophan]] | [[Glycine]]* |- | [[Valine]] | [[Proline]]* |- | | [[Selenocysteine]]* |- | | [[Serine]]* |- | | [[Tyrosine]]* |- | | [[Arginine]]* |- | | [[Histidine]]* |- | | [[Ornithine]]* |- | | [[Taurine]]* |} '''Polar and non polar amino acids and their single and three letter code''' {| class="wikitable sortable" ! Amino Acid ! Three Letter code ! Single Letter code ! Side chain polarity ! Side chain charge (pH 7.4) ! [[Hydropathy index]] ! [[Absorbance]] λ_max(nm) ! ε at λ_max (x10⁻³ M⁻¹ cm⁻¹) |- align="center" | [[Alanine]] | Ala | A | nonpolar | neutral | 1.8 | | |- align="center" | [[Arginine]] | Arg | R | polar | positive | −4.5 | | |- align="center" | [[Asparagine]] | Asn | N | polar | neutral | −3.5 | | |- align="center" | [[Aspartic acid]] | Asp | D | polar | negative | −3.5 | | |- align="center" | [[Cysteine]] | Cys | C | nonpolar | neutral | 2.5 | 250 | 0.3 |- align="center" | [[Glutamic acid]] | Glu | E | polar | negative | −3.5 | | |- align="center" | [[Glutamine]] | Gln | Q | polar | neutral | −3.5 | | |- align="center" | [[Glycine]] | Gly | G | nonpolar | neutral | −0.4 | | |- align="center" | [[Histidine]] | His | H | polar | positive(10%) neutral(90%) | −3.2 | 211 | 5.9 |- align="center" | [[Isoleucine]] | Ile | I | nonpolar | neutral | 4.5 | | |- align="center" | [[Leucine]] | Leu | L | nonpolar | neutral | 3.8 | | |- align="center" | [[Lysine]] | Lys | K | polar | positive | −3.9 | | |- align="center" | [[Methionine]] | Met | M | nonpolar | neutral | 1.9 | | |- align="center" | [[Phenylalanine]] | Phe | F | nonpolar | neutral | 2.8 | 257, 206, 188 | 0.2, 9.3, 60.0 |- align="center" | [[Proline]] | Pro | P | nonpolar | neutral | −1.6 | | |- align="center" | [[Serine]] | Ser | S | polar | neutral | −0.8 | | |- align="center" | [[Threonine]] | Thr | T | polar | neutral | −0.7 | | |- align="center" | [[Tryptophan]] | Trp | W | nonpolar | neutral | −0.9 | 280, 219 | 5.6, 47.0 |- align="center" | [[Tyrosine]] | Tyr | Y | polar | neutral | −1.3 | 274, 222, 193 | 1.4, 8.0, 48.0 |- align="center" | [[Valine]] | Val | V | nonpolar | neutral | 4.2 | | |} Additionally, there are two additional amino acids which are incorporated by overriding stop codons: {| class="wikitable" ! 21st and 22nd amino acids ! 3-Letter ! 1-Letter |- align="center" | [[Selenocysteine]] | Sec | U |- align="center" | [[Pyrrolysine]] | Pyl | O |} In addition to the specific amino acid codes, placeholders are used in cases where [[Protein sequencing|chemical]] or [[X-ray crystallography|crystallographic]] analysis of a peptide or protein can not conclusively determine the identity of a residue. {| class="wikitable" ! Ambiguous Amino Acids ! 3-Letter ! 1-Letter |- align="center" | Asparagine or aspartic acid | Asx | B |- align="center" | Glutamine or glutamic acid | Glx | Z |- align="center" | Leucine or Isoleucine | Xle | J |- align="center" | Unspecified or unknown amino acid | Xaa | X |} ==Amino Acid Structure Gallery== <gallery> image:L-alanine-skeletal.png|[[Alanine|<small>L</small>-Alanine]]<br>(Ala&nbsp;/&nbsp;A) image:L-arginine-skeletal-(tall).png|[[Arginine|<small>L</small>-Arginine]]<br>(Arg&nbsp;/&nbsp;R) image:L-asparagine-skeletal.png|[[Asparagine|<small>L</small>-Asparagine]]<br>(Asn&nbsp;/&nbsp;N) image:L-aspartic-acid-skeletal.png|[[Aspartic acid|<small>L</small>-Aspartic acid]]<br>(Asp&nbsp;/&nbsp;D) image:L-cysteine-skeletal.png|[[Cysteine|<small>L</small>-Cysteine]]<br>(Cys&nbsp;/&nbsp;C) image:Kwas glutaminowy.svg|[[Glutamic acid|<small>L</small>-Glutamic acid]]<br>(Glu&nbsp;/&nbsp;E) image:L-glutamine-skeletal.png|[[Glutamine|<small>L</small>-Glutamine]]<br>(Gln&nbsp;/&nbsp;Q) image:Glycine-skeletal.png|[[Glycine]]<br>(Gly&nbsp;/&nbsp;G) image:L-histidine-skeletal.png|[[Histidine|<small>L</small>-Histidine]]<br>(His&nbsp;/&nbsp;H) image:L-isoleucine-skeletal.svg|[[Isoleucine|<small>L</small>-Isoleucine]]<br>(Ile&nbsp;/&nbsp;I) image:L-leucine-skeletal.svg|[[Leucine|<small>L</small>-Leucine]]<br>(Leu&nbsp;/&nbsp;L) image:L-lysine-skeletal.svg|[[Lysine|<small>L</small>-Lysine]]<br>(Lys&nbsp;/&nbsp;K) image:L-methionine-skeletal.png|[[Methionine|<small>L</small>-Methionine]]<br>(Met&nbsp;/&nbsp;M) image:L-phenylalanine-skeletal.png|[[Phenylalanine|<small>L</small>-Phenylalanine]]<br>(Phe&nbsp;/&nbsp;F) image:Amminoacido prolina formula.svg|[[Proline|<small>L</small>-Proline]]<br>(Pro&nbsp;/&nbsp;P) image:L-serine-skeletal.svg|[[Serine|<small>L</small>-Serine]]<br>(Ser&nbsp;/&nbsp;S) image:L-threonine-skeletal.png|[[Threonine|<small>L</small>-Threonine]]<br>(Thr&nbsp;/&nbsp;T) image:L-tryptophan-skeletal.png|[[Tryptophan|<small>L</small>-Tryptophan]]<br>(Trp&nbsp;/&nbsp;W) image:L-tyrosine-skeletal.png|[[Tyrosine|<small>L</small>-Tyrosine]]<br>(Tyr&nbsp;/&nbsp;Y) image:L-valine-skeletal.png|[[Valine|<small>L</small>-Valine]]<br>(Val&nbsp;/&nbsp;V) image:L-selenocysteine-2D-skeletal.png|[[Selenocysteine|<small>L</small>-Selenocysteine]]<br>(Sec&nbsp;/&nbsp;U) image:Pyrrolysine.svg|[[Pyrrolysine|<small>L</small>-Pyrrolysine]]<br>(Pyl&nbsp;/&nbsp;O) </gallery> The 20 naturally occurring amino acids have different physical and chemical properties, including their electrostatic charge, pKa, hydrophobicity, size and specific functional groups. These properties play a major role in molding protein structure. The salient features of amino acids are described below in the table. {| class="wikitable" |- align="center" ! Amino Acid ! colspan="2" | Abbrev. ! Remarks |- ! [[Alanine]][[Image:L-alanine-3D-balls.png|thumb|170px]] | A | Ala | Very abundant, very versatile. More stiff than glycine, but small enough to pose only small steric limits for the protein conformation. It behaves fairly neutrally, and can be located in both hydrophilic regions on the protein outside and the hydrophobic areas inside. |- ! [[Asparagine]] or [[aspartic acid]] | B | Asx | A placeholder when either amino acid may occupy a position. |- ! [[Cysteine]][[Image:L-cysteine-3D-balls2.png|thumb|170px]] | C | Cys | The sulfur atom bonds readily to [[heavy metals|heavy metal]] ions. Under oxidizing conditions, two cysteines can join together in a [[disulfide bond]] to form the amino acid [[cystine]]. When cystines are part of a protein, [[insulin]] for example, the [[tertiary structure]] is stabilized, which makes the protein more resistant to [[denaturation (biochemistry)|denaturation]]; therefore, disulfide bonds are common in proteins that have to function in harsh environments including digestive enzymes (e.g., [[pepsin]] and [[chymotrypsin]]) and structural proteins (e.g., [[keratin]]). Disulfides are also found in peptides too small to hold a stable shape on their own (eg. [[insulin]]). |- ! [[Aspartic acid]][[Image:L-aspartic-acid-3D-balls.png|thumb|170px]] | D | Asp | Behaves similarly to glutamic acid. Carries a hydrophilic acidic group with strong negative charge. Usually is located on the outer surface of the protein, making it water-soluble. Binds to positively-charged molecules and ions, often used in enzymes to fix the metal ion. When located inside of the protein, aspartate and glutamate are usually paired with arginine and lysine. |- ! [[Glutamic acid]][[Image:L-glutamic-acid-3D-sticks2.png|thumb|170px]] | E | Glu | Behaves similar to aspartic acid. Has longer, slightly more flexible side chain. |- ! [[Phenylalanine]][[ Image: L-phenylalanine-3D-balls.png|thumb|170px]] | F | Phe | [[essential amino acid|Essential]] for humans. Phenylalanine, tyrosine, and tryptophan contain large rigid [[aromaticity|aromatic]] group on the side-chain. These are the biggest amino acids. Like isoleucine, leucine and valine, these are hydrophobic and tend to orient towards the interior of the folded protein molecule. Phenylalanine can be converted into Tyrosine. |- ! [[Glycine]][[Image:Glycine-3D-balls.png|thumb|170px]] | G | Gly | Because of the two hydrogen atoms at the α carbon, glycine is not [[optical isomerism|optically active]]. It is the smallest amino acid, rotates easily, adds flexibility to the protein chain. It is able to fit into the tightest spaces, e.g., the triple helix of [[collagen]]. As too much flexibility is usually not desired, as a structural component it is less common than alanine. |- ! [[Histidine]][[Image:L-histidine-zwitterion-from-xtal-1993-3D-balls-B.png|thumb|170px]] | H | His | In even slightly acidic conditions [[protonation]] of the nitrogen occurs, changing the properties of histidine and the polypeptide as a whole. It is used by many proteins as a regulatory mechanism, changing the conformation and behavior of the polypeptide in acidic regions such as the late [[endosome]] or [[lysosome]], enforcing conformation change in enzymes. However only a few histidines are needed for this, so it is comparatively scarce. |- ! [[Isoleucine]][[Image:L-isoleucine-3D-balls.png|thumb|170px]] | I | Ile | [[essential amino acid|Essential]] for humans. Isoleucine, leucine and valine have large aliphatic hydrophobic side chains. Their molecules are rigid, and their mutual hydrophobic interactions are important for the correct folding of proteins, as these chains tend to be located inside of the protein molecule. |- ! [[Leucine]] or [[isoleucine]] | J | Xle | A placeholder when either amino acid may occupy a position |- ! [[Lysine]][[Image:L-lysine-zwitterion-from-hydrochloride-dihydrate-xtal-3D-balls.png|thumb|170px]] | K | Lys | [[essential amino acid|Essential]] for humans. Behaves similarly to arginine. Contains a long flexible side-chain with a positively-charged end. The flexibility of the chain makes lysine and arginine suitable for binding to molecules with many negative charges on their surfaces. E.g., [[Deoxyribonucleic acids|DNA]]-binding proteins have their active regions rich with arginine and lysine. The strong charge makes these two amino acids prone to be located on the outer hydrophilic surfaces of the proteins; when they are found inside, they are usually paired with a corresponding negatively-charged amino acid, e.g., aspartate or glutamate. |- ! [[Leucine]][[Image:L-leucine-3D-balls.png|thumb|170px]] | L | Leu | [[essential amino acid|Essential]] for humans. Behaves similar to isoleucine and valine. See isoleucine. |- ! [[Methionine]][[Image: L-methionine-B-3D-balls.png|thumb|170px]] | M | Met | [[essential amino acid|Essential]] for humans. Always the first amino acid to be incorporated into a protein; sometimes removed after translation. Like cysteine, contains sulfur, but with a [[methyl]] group instead of hydrogen. This methyl group can be activated, and is used in many reactions where a new carbon atom is being added to another molecule. |- ! [[Asparagine]][[Image: L-asparagine-3D-balls.png|thumb|170px]] | N | Asn | Similar to aspartic acid. Asn contains an [[amide]] group where Asp has a [[carboxyl]]. |- ! [[Pyrrolysine]] | O | Pyl | Similar to [[lysine]], with a [[pyrroline]] ring attached. |- ! [[Proline]][[Image: L-proline-3D-balls.png|thumb|170px]] | P | Pro | Contains an unusual ring to the N-end amine group, which forces the CO-NH amide sequence into a fixed conformation. Can disrupt protein folding structures like [[alpha helix|α helix]] or [[beta sheet|β sheet]], forcing the desired kink in the protein chain. Common in [[collagen]], where it often undergoes a [[posttranslational modification]] to [[hydroxyproline]]. |- ! [[Glutamine]][[Image:L-glutamine-3D-sticks.png|thumb|170px]] | Q | Gln | Similar to glutamic acid. Gln contains an [[amide]] group where Glu has a [[carboxyl]]. Used in proteins and as a storage for [[ammonia]]. The most abundant Amino Acid in the body. |- ! [[Arginine]][[ Image: L-arginine-3D-hztl.png|thumb|170px]] | R | Arg | Functionally similar to lysine. |- ! [[Serine]][[Image:L-serine-3D-balls.png|thumb|170px]] | S | Ser | Serine and threonine have a short group ended with a [[hydroxyl]] group. Its hydrogen is easy to remove, so serine and threonine often act as hydrogen donors in enzymes. Both are very hydrophilic, therefore the outer regions of soluble proteins tend to be rich with them. |- ! [[Threonine]][[Image: L-Threonine-3D-balls.png|thumb|170px]] | T | Thr | [[essential amino acid|Essential]] for humans. Behaves similarly to serine. |- ! [[Selenocysteine]] | U | Sec | [[Selenium|Selenated]] form of cysteine, which replaces [[sulfur]]. |- ! [[Valine]][[Image:L-valine-3D-balls.png|thumb|170px]] | V | Val | [[essential amino acid|Essential]] for humans. Behaves similarly to isoleucine and leucine. See isoleucine. |- ! [[Tryptophan]][[Image: L-tryptophan-3D-balls.png|thumb|170px]] | W | Trp | [[essential amino acid|Essential]] for humans. Behaves similarly to phenylalanine and tyrosine (see phenylalanine). Precursor of [[serotonin]]. Naturally [[fluorescent]]. |- ! Unknown | X | Xaa | Placeholder when the amino acid is unknown or unimportant. |- ! [[Tyrosine]][[Image: L-tyrosine-3D-sticks.png|thumb|170px]] | Y | Tyr | Behaves similarly to phenylalanine (precursor to Tyrosine) and tryptophan (see phenylalanine). Precursor of [[melanin]], [[epinephrine]], and [[thyroid hormone]]s. Naturally [[fluorescent]], although fluorescence is usually quenched by energy transfer to tryptophans. |- ! [[Glutamic acid]] or [[glutamine]] | Z | Glx | A placeholder when either amino acid may occupy a position. |} ==References== {{Reflist|2}} ↑ http://en.wikipedia.org/w/index.php?title=Protein&oldid=425576197 ↑ http://en.wikipedia.org/w/index.php?title=Proteinogenic_amino_acid&oldid=420804587 ↑ http://en.wikipedia.org/w/index.php?title=Amino_acid&oldid=425389108 ↑ http://en.wikipedia.org/w/index.php?title=Amino_acid&oldid=425389108 ↑ Photo-leucine and photo-methionine allow identification of protein-protein interactions in living cells.Nature Methods:4,261–7,2005 ↑ http://en.wikipedia.org/w/index.php?title=Peptide_bond&oldid=417601014 ↑ Basler B, Schuster O, Bach T (November 2005). "Conformationally constrained β-amino acid derivatives by intramolecular [2 + 2]-photocycloaddition of a tetronic acid amide and subsequent lactone ring opening". J. Org. Chem. 70 (24): 9798–808. doi:10.1021/jo0515226. {{PMID|16292808}}. ↑ Murray JK, Farooqi B, Sadowsky JD, et al. (September 2005). "Efficient synthesis of a β-peptide combinatorial library with microwave irradiation". J. Am. Chem. Soc. 127 (38): 13271–80. doi:10.1021/ja052733v. {{PMID|16173757}}. ↑ Seebach D, Matthews JL (1997). "β-Peptides: a surprise at every turn". Chem. Commun. (21): 2015–22. doi:10.1039/a704933a. ↑ http://en.wikipedia.org/w/index.php?title=Enzyme&oldid=424282616 ↑ http://en.wikipedia.org/w/index.php?title=Enzyme&oldid=424282616 ↑ Moss, G.P.. "Recommendations of the Nomenclature Committee". International Union of Biochemistry and Molecular Biology on the Nomenclature and Classification of Enzymes by the Reactions they Catalyse. Retrieved 2006-03-14. ↑ http://en.wikipedia.org/w/index.php?title=Enzyme&oldid=424282616 ↑ Lovell SC et al. (2003). "Structure validation by Cα geometry: φ,ψ and Cβ deviation". Proteins 50 (3): 437–450. doi:10.1002/prot.10286. ... ↑ Crowfoot Hodgkin D (1935). "X-ray Single Crystal Photographs of Insulin". Nature 135: 591. doi:10.1038/135591a0. ↑ Kendrew J. C. et al. (1958-03-08). "A Three-Dimensional Model of the Myoglobin Molecule Obtained by X-Ray Analysis". Nature 181 (4610): 662. doi:10.1038/181662a0. {{PMID|13517261}}. ↑ "Table of entries in the PDB, arranged by experimental method". ↑ "PDB Statistics". RCSB Protein Data Bank. Retrieved 2010-02-09. ↑ Scapin G (2006). "Structural biology and drug discovery". Curr. Pharm. Des. 12 (17): 2087. doi:10.2174/138161206777585201. {{PMID|16796557}}. ↑ Lundstrom K (2006). "Structural genomics for membrane proteins". Cell. Mol. Life Sci. 63 (22): 2597. doi:10.1007/s00018-006-6252-y. {{PMID|17013556}}. ↑ Lundstrom K (2004). "Structural genomics on membrane proteins: mini review". Comb. Chem. High Throughput Screen. 7 (5): 431. {{PMID|15320710}}. ==See also== *[[Biophysics|Department of Biophysics]] **[http://fr.wikiversity.org/wiki/Département:Biophysique '''Departement: Biophysique'''] *[[Nuclear medicine|Department of Nuclear Medicine]] **[http://fr.wikiversity.org/wiki/Département:Médecine_nucléaire Departement:Medecine Nucleaire] *[[Portal:Life Sciences]] *[[Portal:Physics and Astronomy|Portal:Physics]] *[[Portal:Structural biology|Topic:Structural biology]] *[[Structural biology]] [[Category:Biochemistry]] [[Category:Biophysics]] [[Category:Molecular biology]] [[Category:Biology learning projects]] [[Category:Resources]] rkwi9q2xycyh5u4infpdyz7ulq2k1bi 0 139384 2806773 2806548 2026-04-27T10:02:31Z Young1lim 21186 /* Adder */ 2806773 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260427.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] rkojmgrwo2c3qgd2wj5vessiwgv95z2 0 171005 2806778 2806552 2026-04-27T10:22:07Z Young1lim 21186 /* Geometric Series Examples */ 2806778 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260427.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] livtu34m1nxjlhgnhbwyqc9p43mtknf 0 228511 2806738 2760180 2026-04-27T03:53:04Z OhanaUnited 18921 add volume 7 2806738 wikitext text/x-wiki <!--This page uses templates. 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It is also called a '''C₁₂₀''', '''dodecaplex''' (short for "dodecahedral complex"), '''hyperdodecahedron''', '''polydodecahedron''', '''hecatonicosachoron''', '''dodecacontachoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hecatonicosahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> The boundary of the 120-cell is composed of 120 dodecahedral [[W:Cell (mathematics)|cells]] with 4 meeting at each vertex. Together they form 720 [[W:Pentagon|pentagonal]] faces, 1200 edges, and 600 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:Regular dodecahedron|regular dodecahedron]], since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the [[600-cell]]. == Geometry == The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|name=4-polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the [[5-cell|5-cell]],{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=''Remark 5.7''|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional [[W:Swiss Army knife|Swiss Army knife]]: it contains one of everything. It is daunting but instructive to study the 120-cell, because it contains examples of ''every'' relationship among ''all'' the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why [[W:John Stillwell|Stillwell]] titled his paper on the 4-polytopes and the history of mathematics<ref>''Mathematics and Its History'', John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}}</ref> of more than 3 dimensions ''The Story of the 120-cell''.{{Sfn|Stillwell|2001}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} ===Cartesian coordinates=== Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen. ==== √8 radius coordinates ==== The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable |- !24 | ({0, 0, ±2, ±2}) | [[24-cell#Great squares|24-cell]] | rowspan=7 | 600-point 120-cell |- !64 | ({±φ, ±φ, ±φ, ±φ⁻²}) | |- !64 | ({±1, ±1, ±1, ±{{radic|5}}<nowiki />}) | |- !64 | ({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²}) | |- !96 | ([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !96 | ([0, ±φ⁻², ±1, ±φ²]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !192 | ([±φ⁻¹, ±1, ±φ, ±2]) | |} where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[W:Golden ratio|golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618. ==== Unit radius coordinates ==== The unit-radius 120-cell has edge length {{Sfrac|1|φ²{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone, Pileio & Levitt|2010|p=1442|loc=Table 3}} are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} in the left column below: {| class="wikitable" style=width:720px |- !rowspan=3|120 !8 |style="white-space: nowrap;"|({±1, 0, 0, 0}) |[[16-cell#Coordinates|16-cell]] | rowspan="2" |[[24-cell#Great hexagons|24-cell]] | rowspan="3" |[[600-cell#Coordinates|600-cell]] | rowspan="10" style="white-space: nowrap;"|120-cell |- !16 |style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2 |[[W:Tesseract#Radial equilateral symmetry|Tesseract]] |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±1, ±φ]) / 2 |colspan=2|[[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !rowspan=7|480 !colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]] !5-point [[5-cell#Coordinates|5-cell]] ![[24-cell#Great squares|24-cell]] ![[600-cell#Coordinates|600-cell]] |- !32 |style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ⁻²]) / {{radic|8}} |rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br> (−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br> (−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4 |rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0}) |rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br> ({±1, ±1, ±1, ±1}) / 2<br> ([0, ±φ⁻¹, ±1, ±φ]) / 2 |- !32 |style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}} |- !32 |style="white-space: nowrap;"|([±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻², ±1, ±φ²]) / {{radic|8}} |- !192 |style="white-space: nowrap;"|([±φ⁻¹, ±1, ±φ, ±2]) / {{radic|8}} |- |colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[W:Quaternion|quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[W:Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone, Pileio & Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone, Pileio & Levitt|2010|loc=Table 3|p=1442}}}} |} The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}} ===Chords=== [[File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also [[W:Clifford parallel|Clifford parallel]]) and they intersect in a single point. In [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotations]], points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a ''helical'' smooth curve called an ''isocline''{{Efn|An '''isocline''' is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a ''straight line'', a [[W:Geodesic|geodesic]]. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are [[W:Link (knot theory)|linked]] and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} To avoid confusion, we always refer to an ''isocline'' as such, and reserve the term ''[[W:Great circle|great circle]]'' for an ordinary great circle in the plane.|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a [[W:SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has 5-cell edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains an irregular great dodecagon compound of several of these great circle polygons in the same central plane, [[#Compound of five 600-cells|illustrated below]].]] {{see also|600-cell#Golden chords}} The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|[[File:Regular_star_figure_6(5,2).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> six of the 120 disjoint regular 5-cells of edge-length {{radic|2.5}} which are inscribed in the 120-cell appear as six pentagrams, the [[5-cell#Boerdijk–Coxeter helix|Clifford polygon of the 5-cell]]. The 30 vertices comprise a Petrie polygon of the 120-cell,{{Efn|name=two coaxial Petrie 30-gons}} with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In [[600-cell#Decagons and pentadecagrams|600-cell § Decagons and pentadecagrams]], see the illustration of [[W:Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, [[5-cell#Geodesics and rotations|pentagonal circuits]] each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two ''disjoint 5-cells'' are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic [[W:SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>2 disjoint [[W:Pentadecagram|pentadecagram]] isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the ''same'' discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}}]]The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and [[5-cell#Geodesics and rotations|its inscribed regular 5-cells' opposing 1200 edges]].{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an [[#Compound of five 600-cells|irregular great dodecagon]], a compound [[#Chords|great circle polygon of the 120-cell]].|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete [[W:Hopf fibration|Hopf fibration]].{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentadecagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=isocline}} of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} and they are occupied by adjacent [[W:Clifford parallel|Clifford parallel]] great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' rotations.{{Efn|name=distinct rotations}}}} They also give the 120-cell a characteristic great circle polygon: an ''irregular'' great hexagon in which three 120-cell edges alternate with three 5-cell edges.{{Efn|name=irregular great hexagon}} The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the [[5-cell#Geodesics and rotations|5-cell]] and the [[W:8-cell|8-cell tesseract]], they form zig-zag [[W:Petrie polygon|Petrie polygon]]s instead.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical ''isocline''{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its ''Clifford polygon''.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[W:Triacontagon|triacontagon]] {30} zig-zag [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[W:Skew polygon|skew]] regular [[W:Triacontagon|triacontagon]] {30}.{{Efn|name=15 distinct chord lengths}} The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel [[#Compound of five 600-cells|{12} great circle polygons]].]] The 120-cell contains 80 distinct [[W:30-gon|30-gon]] Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons.{{Efn|name=Petrie polygons of the 120-cell}} The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.{{Efn|name=two coaxial Petrie 30-gons}} The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has at least 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the [[W:Petrie polygon|Petrie polygon]] can be picked out as a [[W:Skew polygon|skew polygon]] of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix ring]]. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered [[#Chords|chords]] of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct [[W:Pythagorean distance|Pythagorean distance]]s through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord is the {30/''n''} polygram chord, which connects two vertices that are ''n'' edge lengths apart on a Petrie polygon of the 120-cell. The 15 major chords lie on great circles in central planes that contain regular and irregular polygons of {4}, {10}, or {12} vertices. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). The 15 ''minor chords'' lie on rectangular {4} great circles and do not occur anywhere except inside the 120-cell. In this article, we refer to the 15 minor chords by reference to their arc-angles, e.g. 41.4~° #3⁺ with length {{radic|0.5}} falls between the #3 and #4 chords.|name=additional 120-cell chords}} The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[W:Snub 24-cell|snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=4-polytopes ordered by size and complexity}} The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Great hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[W:Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Great hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[W:H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[W:Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column ''a'' for (30−''i'')₀) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}} |- !colspan=6|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}} ![[5-cell|5-cell]] ![[16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[24-cell|24-cell]] ![[W:Snub 24-cell|Snub]] ![[600-cell]] ![[#Tetrahedrally diminished 120-cell|Dimin]] ! style="border-right: none;"|120-cell ! style="border-left: none;"| |- !colspan=6|Vertices | style="background: seashell;"|5 | style="background: paleturquoise;"|8 | style="background: paleturquoise;"|16 | style="background: paleturquoise;"|24 | style="background: yellow;"|96 | style="background: yellow;"|120 | style="background: seashell;"|480 | style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}} |rowspan=6 style="background: seashell; border: none;"| |- !colspan=6|Edges | style="background: seashell;"|10{{Efn|name=irregular great hexagon}} | style="background: paleturquoise;"|24 | style="background: paleturquoise;"|32 | style="background: paleturquoise;"|96 | style="background: yellow;"|432 | style="background: yellow;"|720 | style="background: seashell;"|1200 | style="background: seashell;"|1200{{Efn|name=irregular great hexagon}} |- !colspan=6|Edge chord | style="background: seashell;{{text color default}};"|#8{{Efn|name=inscribed 5-cells}} | style="background: paleturquoise;"|#7 | style="background: paleturquoise;"|#5 | style="background: paleturquoise;"|#5 | style="background: yellow;"|#3 | style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the [[600-cell]] and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-cell#Boerdijk–Coxeter helix rings|600-cell Petrie polygon is a helical ring]] which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a [[W:Triacontagon#Triacontagram|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a [[600-cell#Decagons|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[600-cell#Boerdijk–Coxeter helix rings|600-cell's Petrie helix]] does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew [[W:Triacontagon#Triacontagram|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[W:Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}} | style="background: seashell;"|#1 | style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}} |- !colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[W:SO(4)#Double rotations|double rotation]] in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its ''rotation angle'' and its ''isocline angle''.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this ''incorrectly'', because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its ''completely'' orthogonal plane, but Clifford parallel to it. It rotates ''with'' its completely orthogonal plane, but not ''in'' it. It is Clifford parallel to its completely orthogonal plane ''and'' to the plane it is moving to, and does not intersect them; the plane that it rotates ''in'' is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, [[16-cell#Rotations|all 6 orthogonal planes]] rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different [[W:Orientation entanglement|orientation]] (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} | style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]] | style="background: paleturquoise;"|[[16-cell#Helical construction|#15]] | style="background: paleturquoise;"|#10 | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]] | style="background: yellow;"|#5 | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]] | style="background: seashell;"|#4 | style="background: seashell;"|#4{{Efn|The characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}} |- !colspan=6|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Efn|name=isoclinic}}|name=Clifford polygon}} | style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]] | style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]] | style="background: paleturquoise;"| | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]] | style="background: yellow;"| | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]] | style="background: seashell;"| | style="background: seashell;"|[[W:Pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}} |- !colspan=3|Chord !Arc !colspan=2|Edge | style="background: seashell;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: yellow;"| | style="background: yellow;"| | style="background: seashell;"| | style="background: seashell;"| |- style="background: seashell;"| |rowspan=2|#1<br>△ |rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]] |rowspan=2|30 |{{Efn|name=120-cell Petrie {30}-gon}} |colspan=2|120-cell edge <big>𝛇</big> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |15.5~° |{{radic|𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <small>{{sfrac|1|φ}}</small> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² = <small>{{sfrac|1|φ²}}</small> ≈ 0.382 {{indent|7}}𝜀 = 𝚫²/2 = <small>{{sfrac|1|2φ⁴}}</small> ≈ 0.073<br> and the 120-cell edge-length is: {{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270<br> For example: {{indent|7}}𝛇 = {{radic|𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}} |0.270~ |- style="background: seashell;"| |rowspan=2|#2<br><big>☐</big> |rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]] |rowspan=2|15 | |colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1₀. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[W:Pentagon#Regular pentagons|pentagon diagonal]] (the #2 chord) is in the [[W:Golden ratio|golden ratio]] φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn|name=dodecahedral cell metrics}}|name=face pentagon chord}}|name=#2 chord}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |25.2~° |{{radic|0.19~}} |0.437~ |- style="background: yellow;"| |rowspan=2|#3<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]] |rowspan=2|10 |𝝅/5 |colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |- style="background: seashell;"| |rowspan=2|#4<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]] |rowspan=2|{{sfrac|15|2}} | |colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |44.5~° |{{radic|0.57~}} |0.757~ |- style="background: paleturquoise;"| |rowspan=2|#5<br>△ |rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]] |rowspan=2|6 |𝝅/3 |colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[W:Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a [[W:0-gon|0-gon]] great circle axis that does not intersect any vertices.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different [[#Compound of five 600-cells|central planes]].|name=great hexagon}} |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2|{{green|<big>'''225'''</big>}}<br><br> |rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |60° |{{radic|1}} |1 |- style="background: yellow;"| |rowspan=2|#6<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]] |rowspan=2|5 |2𝝅/5 |colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |- style="background: paleturquoise;"| |rowspan=2|#7<br><big>☐</big> |rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]] |rowspan=2|{{sfrac|30|7}} |𝝅/2 |colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}} |rowspan=2| |rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24 |rowspan=2|{{green|<big>'''675'''</big>}}<br>48 |rowspan=2|<br>72 |rowspan=2| |rowspan=2|<br>1800 |rowspan=2|<br> |rowspan=2|<br>9000 |rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4} |- style="background: paleturquoise;"| |90° |{{radic|2}} |1.414~ |- style="background: #FFCCCC;"| |rowspan=2|#8<br><big>✩</big> |rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]] |rowspan=2|{{sfrac|15|4}} | |colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[W:Triacontagon|triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}} |rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10 |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: #FFCCCC;"| |104.5~° |{{radic|2.5}} |1.581~ |- style="background: yellow;"| |rowspan=2|#9<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]] |rowspan=2|{{sfrac|10|3}} |3𝝅/5 |colspan=2|[[W:Golden section|golden section]] <math>\phi</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |108° |{{radic|2.𝚽}} |1.618~ |- style="background: paleturquoise;"| |rowspan=2|#10<br>△ |rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]] |rowspan=2|3 |2𝝅/3 |colspan=2|[[24-cell#Great triangles|great triangle]] |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2| |rowspan=2|<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |120° |{{radic|3}} |1.732~ |- style="background: seashell;"| |rowspan=2|#11<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]] |rowspan=2|{{sfrac|30|11}} | |colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: yellow;"| |rowspan=2|#12<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]] |rowspan=2|{{sfrac|5|2}} |4𝝅/5 |colspan=2|great [[W:Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: seashell;"| |rowspan=2|#13<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]] |rowspan=2|{{sfrac|30|13}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/13}-gram]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: seashell;"| |rowspan=2|#14<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]] |rowspan=2|{{sfrac|15|7}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/14}=2{15/7}]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200<br> |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: paleturquoise;"| |rowspan=2|#15<br><small>△☐𝜙</small> |rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]] |rowspan=2|2 |𝝅 |colspan=2|[[W:Diameter|diameter]] |rowspan=2| |rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4 |rowspan=2|<br>8 |rowspan=2|<br>12 |rowspan=2|<br>48 |rowspan=2|<br>60 |rowspan=2|<br>240 |rowspan=2|<br>300{{Efn|name=rays and bases}} |rowspan=2|{{blue|<big>'''1'''</big>}}<br><br> |- style="background: paleturquoise;"| |180° |{{radic|4}} |2 |- !colspan=6|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} | style="background: seashell;"|25 | style="background: paleturquoise;"|64 | style="background: paleturquoise;"|256 | style="background: paleturquoise;"|576 | style="background: yellow;"| | style="background: yellow;"|14400 | style="background: seashell;"| | style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}} !<big>{{blue|'''300'''}}</big> |} [[File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.{{Efn|Drawing the fan of chords with #1 and #11 at a different origin than all the others is an artistic choice, since all the chords are incident at every vertex. We could just as well have drawn all the chords from the same origin vertex, but this arrangement notices the parallel relationship between #8 and #11.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] The annotated chord table is a complete [[W:Bill of materials|bill of materials]] for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the '''#3''' chord row, the 600-cell's 72 great decagons contain 720 '''#3''' chords in all. The '''{{red|red}}''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of <big>{{red|'''25'''}}</big> disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''{{green|green}}''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains <big>{{green|'''225'''}}</big> distinct 24-cells which share components. The '''{{blue|blue}}''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the '''#3''' chord row, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[W:Vertex figure|vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell. === Relationships among interior polytopes === The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, ''except'' for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which ''does'' occur. Various regular [[W:Skew polygon|skew polygon]]s {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional [[W:Jigsaw puzzle|jigsaw puzzle]] in which all those polytopes are the parts.{{Sfn|Schleimer & Segerman|2013}} Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.{{Sfn|Stillwell|2001}} The regular 1-polytope occurs in only [[#Chords|15 distinct lengths]] in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By [[W:Alexandrov's uniqueness theorem|Alexandrov's uniqueness theorem]], convex polyhedra with shapes distinct from each other also have distinct [[W:Metric spaces|metric spaces]] of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four {{background color|paleturquoise|[[24-cell#Hypercubic chords|hypercubic chords]]}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built solely from them. {{see also|W:24-cell#Relationships among interior polytopes|label 1=24-cell § Relationships among interior polytopes}} An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|[[600-cell#Golden chords|golden chords]]}} are square roots of irrational fractions that are functions of {{radic|5}}. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built solely from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. {{see also|W:600-cell#Icosahedra|label 1=600-cell § Icosahedra}} All 15 major chords, and 15 other distinct chordal distances (the minor chords [[120-cell#Geodesic rectangles|enumerated below]]), occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{background color|#FFCCCC|[[5-cell#Boerdijk–Coxeter helix|regular 5-cells and {{radic|5/2}} chords]].}}{{Efn|Dodecahedra emerge as ''visible'' features in the 120-cell, but they also occur in the 600-cell as ''interior'' polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the ''regular'' 5-cell (the [[W:Simplex|simplex]] regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the [[W:Tetrahedron|3-simplex]] and [[W:5-orthoplex|5-orthoplex]]. An [[W:simplex|<math>n</math>-simplex]] is bounded by <math>n+1</math> vertices and <math>n+1</math> (<math>n</math>-1)-simplex facets, and has <math>z+1</math> long diameters (its edges) of length <math>\sqrt{n+1}/\sqrt{n}</math> radii. An [[W:orthoplex|<math>n</math>-orthoplex]] is bounded by <math>2n</math> vertices and <math>2^n</math> (<math>n</math>-1)-simplex facets, and has <math>n</math> long diameters (its orthogonal axes) of length <math>2</math> radii. An [[W:hypercube|<math>n</math>-cube]] is bounded by <math>2^n</math> vertices and <math>2n</math> (<math>n</math>-1)-cube facets, and has <math>2^{n-1}</math> long diameters of length <math>\sqrt{n}</math> radii.{{Efn|The <math>n</math>-simplex's facets are larger than the <math>n</math>-orthoplex's facets. For <math>n=4</math>, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of <math>\sqrt{5}</math> to <math>\sqrt{4}</math> to <math>\sqrt{2}</math>.|name=root 5/root 4/root 2}} The <math>\sqrt{3}</math> long diameters of the 3-cube are shorter than the <math>\sqrt{4}</math> axes of the 3-orthoplex. The [[16-cell#Coordinates|coordinates of the 4-orthoplex]] are the permutations of <math>(0,0,0,\pm 1)</math>, and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of <math>(0,0,0,1)</math>.{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting <math>(0,0,0,1)</math>, and its completely orthogonal 3-facet permuting <math>(0,0,0,-1)</math>, comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-[[W:demihypercube|demicube]], half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to <math>n=4</math>. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} The <math>\sqrt{4}</math> long diameters of the 4-cube are the same length as the <math>\sqrt{4}</math> axes of the 4-orthoplex. The [[W:5-orthoplex#Cartesian coordinates|coordinates of the 5-orthoplex]] are the permutations of <math>(0,0,0,0,\pm 1)</math>, and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of <math>(0,0,0,0,1)</math>.{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting <math>(0,0,0,0,1)</math>, and its completely orthogonal 4-facet permuting <math>(0,0,0,0,-1)</math>, comprise all 10 vertices of the 5-orthoplex.}} The <math>\sqrt{5}</math> long diameters of the 5-cube are longer than the <math>\sqrt{4}</math> axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells.{{Efn|No vertex pair of any of the 120 5-cells (no [[5-cell#Geodesics and rotations|great digon central plane of a 5-cell]]) occurs in any of the 675 16-cells (the 675 [[16-cell#Coordinates|Cartesian basis sets of 6 orthogonal central planes]]).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}} {{see also|W:5-cell#Geodesics and rotations|label 1=5-cell § Geodesics and rotations}} === Compound of five 600-cells === [[File:Great dodecagon of the 120-cell.png|thumb|300px|The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).]] The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} As a consequence of being a compound of five disjoint 600-cells, the 120-cell has 200 irregular great dodecagon {12} central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 {12} central planes originate as the compounds of the hexagonal central planes of the 25 disjoint inscribed 24-cells and the digon central planes of the 120 disjoint inscribed regular 5-cells; they contain all the 24-cell and 5-cell edges, and also the 120-cell edges. Thus the edges and characteristic rotations{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The '''characteristic isoclinic rotation of a 4-polytope''' is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each [[W:Hopf fibration|Hopf fibration]] of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an [[#Compound of five 600-cells|irregular great dodecagon]] which also has #4 chord edges. In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in these same 200 rotation planes.{{Efn|name=edge rotation planes}} Each of the ten 600-cells occupies the entire set of 200 planes. The 120-cell's irregular [[#Other great circle constructs|dodecagon {12} great circle polygon]] has 6 short edges (#1 [[#Chords|chords]] marked {{Color|red|𝜁}}) alternating with 6 longer dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} Inscribed in the irregular great dodecagon are two irregular great hexagons ({{color|red|red}}) in alternate positions.{{Efn|name=irregular great hexagon}} Two ''regular'' great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of 200 dodecagon planes. They occur as 100 completely orthogonal pairs, and they are the ''same'' 200 central planes each containing a [[600-cell#Hexagons|hexagon]] that are found in ''each'' of the 10 inscribed 600-cells. There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares regular hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint regular hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells. ===Concentric hulls=== [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px| Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an Overall Hull that is a [[W:Chamfered dodecahedron|chamfered dodecahedron]] of Norm={{radic|8}}.<br /> Hulls 1 - 8 are the 8 sections of the 120-cell beginning with a cell (Hull 1).<br /> Hulls 1, 2, & 7 are each pairs of [[W:Dodecahedron|dodecahedron]]s.<br /> Hull 3 is a pair of [[W:Icosidodecahedron|icosidodecahedron]]s.<br /> Hulls 4 & 5 are each pairs of [[W:Truncated icosahedron|truncated icosahedron]]s.<br /> Hull 6 is a pair of semi-regular [[W:Rhombicosidodecahedron|rhombicosidodecahedron]]s.<br /> Hull 8 is a single non-uniform [[W:Rhombicosidodecahedron#Names|rhombicosidodecahedron]], the central section.<br /> A more detailed visualization of these 15 simplified sections, with subgroup sections where the inscribed solid has more than one permutation in its orbit, is available [https://commons.wikimedia.org/wiki/File:Cell_First_533_120-Cell_Sections.svg here].]] {{Clear}} These hulls illustrate Coxeter's sections 1₃ - 8₃ of the 120-cell, the sections beginning with a cell (hull #1).{{Sfn|Coxeter|1973|p=299|loc=Table V (iv) Sections of {5,3,3} beginning with a cell (right half of table)}} A ''section'' is a flat 3-dimensional hyperplane slice through the [[W:3-sphere|3-sphere]]: a 2-sphere (ordinary sphere). It is dimensionally analogous to a flat 2-dimensional plane slice through a 2-sphere: a 1-sphere (ordinary circle). The hulls are illustrated as if they were all the same size, but actually they increase in radius as numbered: they are concentric 2-spheres that nest inside each other. Every cell of the 120-cell is the smallest hull in its own set of 8 concentric hulls. There are 120 distinct sets of hulls. The 120-cell actually has 15 sections beginning with a cell, numbered 1 - 15 with number 8 in the center. After increasing in size from 1 to 8, the hulls get smaller again. Sections 1 and 15 are both a hull #1, the smallest hull, a dodecahedral cell of the 120-cell. Section 8 is the central section, the largest hull, with the same radius as the 120-cell. Except for the central section 8, the sections occur in parallel pairs, on either side of the central section. Hull #8 is dimensionally analogous to the equator, while hulls #1 - #7 are dimensionally analogous to lines of latitude. There are 120 of each kind of hull #1 - #7 in the 120-cell, but only 60 of the central hull #8. {{Clear}} The 120-cell also has 30 sections beginning with a vertex, illustrated below. Like the sections beginning with a cell illustrated above, the vertex-first sections are also flat 3-dimensional hyperplane slices through the 3-sphere, polyhedra that nest inside each other as concentric 2-spheres. Section 0₀ is the vertex itself. Section 1₀ is the 120-cell's tetrahedral vertex figure. Sections 1₀ - 29₀ are described in more detail in [[120-cell#Geodesic rectangles|§Geodesic rectangles]] below. {{Clear}} [[File:Vertex_First_533_120-Cell_Sections.svg|thumb|left|640px| Coxeter's sections 0₀ - 30₀ of the 120-cell, the sections beginning with a vertex, showing the orbit sections and subgroup sections (when the inscribed solid has more than one permutation in its orbit), as well as the convex hull of each orbit on the right.]] {{Clear}} === Geodesic rectangles === The 30 distinct chords{{Efn|name=additional 120-cell chords}} found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Background color|palegreen|△ planes that intersect {12} vertices}} in an [[#Compound of five 600-cells|irregular great dodecagon]], {{Background color|yellow|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and <big>☐</big> planes that intersect {4} vertices in several kinds of {{Background color|gainsboro|rectangle}}, including a {{Background color|seashell|square}}. Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing [[#Concentric hulls|polyhedral sections]] of the 120-cell beginning with a vertex (the 0₀ section), as illustrated above. The correspondence is that each 120-cell vertex is surrounded in curved 3-space <math>S_3</math> by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius in Euclidean 3-space <math>R_3</math>.{{Efn|In the curved 3-dimensional space <math>S_3</math> of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the ''apex'' vertex and its surrounding ''base'' polyhedron form a [[W:Polyhedral pyramid|polyhedral pyramid]]. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} There are 600 distinct sets of 15 hulls. The #1 chord is the radius in <math>S_3</math> of the 1₀ section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the radius in <math>S_3</math> of its congruent opposing 29₀ section. The #7 chord is the radius in <math>S_3</math> of the central vertex-first section of the 120-cell, in which two opposing 15₀ sections are coincident. Each vertex is surrounded by two instances of each polyhedron, at the near and far radial distances of the polyhedron's 180° complementary chords, but because curved space <math>S_3</math> begins to close back up on itself after the #7 90° chord, the near and far concentric polyhedra are the same size. Each chord length is given three ways (on successive lines): for the unit-radius 120-cell as a square root, for the unit-radius 120-cell, and for the unit-edge 120-cell.{{Efn|We give chord lengths as unit-radius square roots in these articles, even when they are integers (e.g. the long diameter is {{radic|4}}). Our usual metric is unit-radius, which reveals relationships among successive 4-polytopes,{{Efn|name=4-polytopes ordered by size and complexity}} but Coxeter{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} and Steinbach{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with "fan of chords" diagrams.|p=23|loc=Figure 3}} use unit-edge, which reveals relationships among successive chords.|name=metrics}} To the left of this last unit-edge metric, its reciprocal^-1 is given. The reciprocal is the long radius of a regular ''n''₀-polygon with unit-radius 120-cell edges (#1 chords) as its edges; but this does not imply that the section ''n''₀ polyhedron contains any ''n''₀ polygons.{{Efn|The 120-cell contains no regular {30} central polygons, although its Petrie polygon is a skew regular {30}. Therefore the edge of the regular triacontagon {30} is not a chord of the 120-cell represented in this table. Nevertheless these metrics of the {30} are relevant:<br> :Unit-radius {30}: ::Edge <small><math>E = 2 \sin{\pi/30} \approx \sqrt{0.0437} \approx 0.209</math></small> :Unit-edge {30}: ::Radius <small><math>R_{ue} = 1/E \approx 4.783</math></small> :{30} with 120-cell edges: ::Edge <small><math>\zeta \approx 0.270~</math></small> ::<small><math>E \approx 0.774 \times \zeta</math></small> ::Radius <small><math>R_\zeta \approx 1.292</math></small> |name=triacontagon metrics}} {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |#0<br><br>0₀ | |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |<math>\pi</math> |{{radic|4}} |{{radic|4}} | rowspan="3" |#15<br><br>30₀ |- style="background: palegreen;" | |0° |0 |0 |180° |2 |2 |- style="background: palegreen;" | | |0 |<small><math>0\times\zeta</math></small> |0.135~^-1 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#1<br><br>1₀ |𝞯 |{{radic|0.𝜀}}{{Efn|name=fractional square roots}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} | |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |#14<br><br>29₀ |- style="background: palegreen;" | |15.5~°{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}} |0.270~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |164.5~° |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |1^-1 |1 |<small><math>1\times\zeta</math></small> |0.136~^-1 |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#2<br><br>2₀ |{{Efn|name=#2 chord}} |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 | |{{radic|3.81~}} | | rowspan="3" |#13<br><br>28₀ |- style="background: gainsboro;" | |25.2~° |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |154.8~° |1.952~ | |- style="background: gainsboro;" | |0.618~^-1 |1.618~ |<small><math>\phi\times\zeta</math></small> |0.138~^-1 |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |#3<br><br>3₀ |<math>\pi / 5</math> |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |<math>4\pi / 5</math> |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |#12<br><br>27₀ |- style="background: yellow;" | |36° |0.618~ |<small><math>1 / \phi</math></small> |144°{{Efn|name=dihedral}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.437~^-1 |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |0.142~^-1 |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#3⁺<br><br>4₀ | |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |#12⁻<br><br>26₀ |- style="background: gainsboro;" | |41.4~° |0.707~ |<small><math>\sqrt{2}/2</math></small> |138.6~° |1.871~ | |- style="background: gainsboro;" | |0.382~^-1 |2.618~ |<small><math>\phi^2\times\zeta</math></small> |0.144~^-1 |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#4<br><br>5₀ | |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown). Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' great hexagons (the [[600-cell#Hexagons|24-cell's characteristic rotation]]), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in ''irregular'' great hexagons).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every ''regular'' great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.|name=#4 isocline chord bridge}}|name=dodecagon rotation}}<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" |{{Efn|name=#4 isocline chord bridge}} | |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |#11<br><br>25₀ |- style="background: palegreen;" | |44.5~° |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |135.5~° |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.357~^-1 |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |0.146~^-1 |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#4⁺<br><br>6₀ | |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |#11⁻<br><br>24₀ |- style="background: gainsboro;" | |49.1~° |0.831~ | |130.9~° |1.819~ | |- style="background: gainsboro;" | |0.325~^-1 |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |0.148~^-1 |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁻<br><br>7₀ | |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |#10⁺<br><br>23₀ |- style="background: gainsboro;" | |56° |0.939~ | |124° |1.766~ | |- style="background: gainsboro;" | |0.288~^-1 |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |0.153~^-1 |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |#5<br><br>8₀ |<math>\pi / 3</math> |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}}<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |<small><math>2\pi / 3</math></small> |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |#10<br><br>22₀ |- style="background: palegreen;" | |60° |1 | |120° |1.732~ | |- style="background: palegreen;" | |0.270~^-1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |0.156~^-1 |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁺<br><br>9₀ | |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |#10⁻<br><br>21₀ |- style="background: gainsboro;" | |66.1~° |1.091~ | |113.9~° |1.676~ | |- style="background: gainsboro;" | |0.247~^-1 |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |0.161~^-1 |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁻<br><br>10₀ | |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |#9⁺<br><br>20₀ |- style="background: gainsboro;" | |69.8~° |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |110.2~° |1.640~ | |- style="background: gainsboro;" | |0.236~^-1 |4.236~ |<small><math>\phi^3\times\zeta</math></small> |0.165~^-1 |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |#6<br><br>11₀ |<math>2\pi/5</math> |{{radic|1.𝚫}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}}<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |<math>3\pi / 5</math> |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |#9<br><br>19₀ |- style="background: yellow;" | |72° |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |108° |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |0.230~^-1 |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |0.167~^-1 |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |#6⁺⁻<br><br>12₀ | |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic|2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12₀ sections and 18₀ sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}}<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 | |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |#8<br><br>18₀ |- style="background: palegreen;" | |75.5~° |1.224~ | |104.5~° |1.581~ | |- style="background: palegreen;" | |0.221~^-1 |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |0.171~^-1 |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺<br><br>13₀ | |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.31~}} | | rowspan="3" |#8⁻<br><br>17₀ |- style="background: gainsboro;" | |81.1~° |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |98.9~° |1.520~ | |- style="background: gainsboro;" | |0.208~⁻¹ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |0.178~^-1 |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺⁺<br><br>14₀ | |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |#7⁺<br><br>16₀ |- style="background: gainsboro;" | |84.5~° |1.345~ | |95.5~° |1.480~ | |- style="background: gainsboro;" | |0.201~⁻¹ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |0.182~^-1 |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |#7<br><br>15₀ |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}}<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |#7<br><br>15₀ |- style="background: seashell;" | |90° |1.414~ | |90° |1.414~ | |- style="background: seashell;" | |0.191~⁻¹ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |0.191~^-1 |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|name=isoclinic rotation}} of a distinct class,{{Efn|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons" / 4.}} In each class of rotation,{{Efn|[[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]] are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are ''invariant'', which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually [[W:Clifford parallel|Clifford parallel]]. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called ''fibrations'', which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular ''fibers'', the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} ===Polyhedral graph=== Considering the [[W:Adjacency matrix|adjacency matrix]] of the vertices representing the polyhedral graph of the unit-radius 120-cell, the [[W:Graph diameter|graph diameter]] is 15, connecting each vertex to its coordinate-negation at a [[W:Euclidean distance|Euclidean distance]] of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are [[W:Vertex coloring|3-colorable]]. The graph is [[W:Eulerian path|Eulerian]] having degree 4 in every vertex. Its edge set can be decomposed into two [[W:Hamiltonian path|Hamiltonian cycles]].<ref>{{cite book| author = Carlo H. Séquin | title = Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes | date = July 2005 | pages = 463–472 | publisher = Mathartfun.com | isbn = 9780966520163 | url = https://archive.bridgesmathart.org/2005/bridges2005-463.html#gsc.tab=0 | access-date=March 13, 2023}}</ref> === Constructions === The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the [[600-cell]],{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the [[24-cell|24-cell]],{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 24-cell}} the 24-cell can be deconstructed into three distinct instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 '''rays''' [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a '''[[W:Orthonormal basis|basis]]'''. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 300₉675₄ to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct [[16-cell#Coordinates|16-cell]] containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse [[W:Golden ratio|golden ratio]]), but the 120-cell's edge length is ~0.270 times its radius. The 120-cell is also the convex hull of the regular compound of 120 disjoint regular 5-cells. This can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point 600-cell, expand each vertex into a regular 5-cell. For each of the 120 vertices, add 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point 600-cells: a 120-cell. ==== Dual 600-cells ==== [[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]] Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ({{sfrac|φ²|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ²{{radic|2}} ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The [[#Unit radius coordinates|unit radius 120-cell]] (with edge-length {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ²}} ≈ 1.080. [[File:Dodecahedron_vertices.svg|thumb|150px|right|One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.{{Efn|In the [[W:120-cell#Dual 600-cells|dodecahedral cell]] of the unit-radius 120-cell, the edge is the '''15.5° #1 [[#Chords|chord]]''' of the 120-cell of length <small><math>\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270</math></small>. Eight {{Color|orange}} vertices lie at the Cartesian coordinates <small><math>(\pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8})</math></small> relative to origin at the cell center. They form a cube (dashed lines) whose edges are the '''25.2° #2 chord''' of length <small><math>\tfrac{1}{\phi\sqrt{2}} \approx 0.437</math></small> (the pentagon diagonal). The face diagonals of the cube (not drawn) are the '''36° #3 chord''' of length <small><math>\tfrac{1}{\phi} \approx 0.618</math></small> (the edges of two 600-cell tetrahedron cells inscribed in the cube). The next largest '''41.1° chord''' has length <small><math>\tfrac{1}{\sqrt{2}} \approx 0.707</math></small>. The diameter of the dodecahedron is the '''44.5° #4 chord''' of length <small><math>\tfrac{\sqrt{3}}{\phi\sqrt{2}} \approx 0.757</math></small> (the cube diagonal). If the #4 diameter is extended outside the dodecahedron in a straight line in the curved space of the 3-sphere, it is colinear with a #1 edge belonging to three neighboring dodecahedron cells, and the combined '''60° #5 chord''' has length <small><math>\sqrt{1}</math></small> (an edge of an inscribed 24-cell). If this 60° combined #4 plus #1 geodesic is further extended in a straight line by another #4 chord (the diameter of a further cell), the combined '''104.5° #8 chord''' has length <small><math>\tfrac{\sqrt{5}}{\sqrt{2}} \approx 1.581</math></small> (an edge of an inscribed regular 5-cell).|name=dodecahedral cell metrics}}]] Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ²|{{radic|8}}}} ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given [[#√8 radius coordinates|above]] of long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ²}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ², which is smaller than {{Radic|8}} in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ²{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps. ==== Cell rotations of inscribed duals ==== Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} This shows that the 120-cell contains, among its many interior features, 120 [[W:Compound of ten tetrahedra|compounds of ten tetrahedra]], each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because [[#Cell rotations of inscribed duals|each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells]]. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 600-cells}} All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-cell.}} Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. ==== Augmentation ==== Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing [[W:Hyperpyramid|4-pyramid]]s of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a [[W:Regular dodecahedron#Cartesian coordinates|dodecahedron]].{{Efn|name=truncated apex}} Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a [[600-cell#Icosahedra|cluster of five tetrahedra]], with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.{{Efn|As we saw in the [[600-cell#Cell clusters|600-cell]], these 12 tetrahedra belong (in pairs) to the 6 [[600-cell#Icosahedra|icosahedral clusters]] of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. ==== Weyl orbits ==== Another construction method uses [[W:Quaternion|quaternion]]s and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe <math>T</math> and <math>T'</math> [[24-cell|24-cell]]s as quaternion orbit weights of D4 under the Weyl group W(D4):<br/> O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}<br/> O(1000) : V1<br/> O(0010) : V2<br/> O(0001) : V3 <math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math> With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the 120-cell of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the [[600-cell]] <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> * the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math> * the [[W:Dual snub 24-cell|dual snub 24-cell]] = <math>T \oplus T' \oplus S'</math>. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}} <math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} ! [[W:K-face|''k''-face]]||f_k||f₀ || f₁||f₂||f₃||[[W:vertex figure|''k''-fig]] !Notes |- align=right |A₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|3|node}} ||( ) !f₀ || 600 || 4 || 6 || 4 ||[[W:Regular tetrahedron|{3,3}]] || H₄/A₃ = 14400/24 = 600 |- align=right |A₁A₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|3|node}} ||{ } !f₁ || 2 || 1200 || 3 || 3 || [[W:Equilateral triangle|{3}]] || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |H₂A₁ ||{{Coxeter–Dynkin diagram|node_1|5|node|2|node_x|2|node}} ||[[W:Pentagon|{5}]] !f₂ || 5 || 5 || 720 || 2 || { } || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |H₃ ||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|2|node_x}} ||[[W:Regular dodecahedron|{5,3}]] !f₃ || 20 || 30 || 12 ||120|| ( ) || H₄/H₃ = 14400/120 = 120 |} == Visualization == The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[24-cell|24-cell]]). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}} === Layered stereographic projection === The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer & Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12). Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an [[W:Icosidodecahedron|icosidodecahedron]], with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="4" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 12 cells | First layer of meridional cells / "[[W:Arctic Circle|Arctic Circle]]" | style="text-align: center" | 36° |- | style="text-align: center" | 3 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 60° |- | style="text-align: center" | 4 | style="text-align: center" | 12 cells | Second layer of meridional cells / "[[W:Tropic of Cancer|Tropic of Cancer]]" | style="text-align: center" | 72° |- | style="text-align: center" | 5 | style="text-align: center" | 30 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" | Equator |- | style="text-align: center" | 6 | style="text-align: center" | 12 cells | Third layer of meridional cells / "[[W:Tropic of Capricorn|Tropic of Capricorn]]" | style="text-align: center" | 108° | rowspan="4" | Southern Hemisphere |- | style="text-align: center" | 7 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 120° |- | style="text-align: center" | 8 | style="text-align: center" | 12 cells | Fourth layer of meridional cells / "[[W:Antarctic Circle|Antarctic Circle]]" | style="text-align: center" | 144° |- | style="text-align: center" | 9 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 120 cells ! colspan="3" | |} The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell. === Intertwining rings === [[Image:120-cell rings.jpg|right|thumb|300px|Two intertwining rings of the 120-cell.]] [[File:120-cell_two_orthogonal_rings.png|thumb|300px|Two orthogonal rings in a cell-centered projection]] The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized [[W:Hopf fibration|Hopf fibration]].{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer & Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical [[W:Torsion of a curve|torsion]]. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two ([[W:Clifford torus|Clifford]]) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B²|pp=23-29}} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint ([[W:Clifford parallel|Clifford parallel]]) great circles. === Other great circle constructs === There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter [[#Chords|chords]], forming an [[#Compound of five 600-cells|irregular dodecagon in a central plane]]. Both these great circle paths have dual [[600-cell#Union of two tori|great circle paths in the 600-cell]]. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a [[600-cell#Decagons|decagon]].{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six [[W:Triangular bipyramids|triangular bipyramids]]) in the 600-cell. This latter path corresponds to a [[600-cell#Icosahedra|ring of six icosahedra]] meeting face to face in the [[W:Snub 24-cell|snub 24-cell]] (or [[W:Icosahedral pyramid|icosahedral pyramids]] in the 600-cell), forming a [[600-cell#Hexagons|hexagon]]. Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges [[#Chords|illustrated above]].{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each [[#Compound of five 600-cells|irregular great dodecagon]], in alternate positions. === 2D Orthogonal projections === [[W:Orthographic projection|Orthogonal projection]]s of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by [[W:B. L. Chilton|B. L. Chilton]].{{Sfn|Chilton|1964}} The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:Van Oss polygon|van Oss polygon]]. {| class="wikitable" |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:120-cell graph H4.svg|240px]]<br>[30]<br>(Red=1) |[[File:120-cell t0 p20.svg|240px]]<br>[20]<br>(Red=1) |[[File:120-cell t0 F4.svg|240px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:120-cell t0 H3.svg|240px]]<br>[10]<br>(Red=5, orange=10) |[[File:120-cell t0 A2.svg|240px]]<br>[6]<br>(Red=1, orange=3, yellow=6, lime=9, green=12) |[[File:120-cell t0 A3.svg|240px]]<br>[4]<br>(Red=1, orange=2, yellow=4, lime=6, green=8) |} === 3D Perspective projections === These projections use [[W:Perspective projection|perspective projection]], from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. A comparison of perspective projections of the 3D dodecahedron to 2D (above left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. [[W:Schlegel diagram|Schlegel diagram]]s use [[W:Perspective (graphical)|perspective]] to show depth in the dimension which has been flattened, choosing a view point ''above'' a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. [[W:Stereographic projection|Stereographic projection]]s use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a [[W:3-sphere|3-sphere]]. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating [[120-cell#Animations|animations]] below, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must). {| class="wikitable" style="width:540px;" |+Comparison with regular dodecahedron |- !width=80|Projection ![[W:Dodecahedron|Dodecahedron]] !120-cell |- ![[W:Schlegel diagram|Schlegel diagram]] |align=center|[[Image:Dodecahedron schlegel.svg|220px]]<br>12 pentagon faces in the plane |align=center|[[File:Schlegel wireframe 120-cell.png|220px]]<br>120 dodecahedral cells in 3-space |- ![[W:Stereographic projection|Stereographic projection]] |align=center|[[Image:Dodecahedron stereographic projection.png|220px]] |align=center|[[Image:Stereographic polytope 120cell faces.png|220px]]<br>With transparent faces |} {|class="wikitable" |- !colspan=2|Enhanced perspective projections |- |align=center|[[Image:120-cell perspective-cell-first-02.png|240px]] |Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: * Nearest dodecahedron to the 4D viewpoint rendered in yellow * The 12 dodecahedra immediately adjoining it rendered in cyan; * The remaining dodecahedra rendered in green; * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image. |- |align=center|[[Image:120-cell perspective-vertex-first-02.png|240px]] |Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: * Four cells surrounding nearest vertex shown in 4 colors * Nearest vertex shown in white (center of image where 4 cells meet) * Remaining cells shown in transparent green * Cells facing away from 4D viewpoint culled for clarity |} === Animations === {|class="wikitable" !colspan=2|Projections to 3D of a 4D 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]] |- |align=center|[[File:120-cell.gif|256px]] |align=center|[[File:120-cell-inner.gif|256px]] |- |From outside the [[W:3-sphere|3-sphere]] in 4-space. |Inside the [[600-cell#Boundary envelopes|3D surface]] of the 3-sphere. |} In all the above projections of the 120-cell, only the edges of the 120-cell appear. All the other [[#Chords|chords]] are not shown. 600 chords converge at ''each'' of the 600 vertices. The complex [[#Relationships among interior polytopes|interior parts]] of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.{{Efn|[[File:Omnitruncated_120-cell_Coxeter_sections-subsections_projected_from_4D.svg|thumb|A full display of each section's orbits along with sub-section orbits in the 14400-point omnitruncated 120-cell.]]The 120-cell has <small><math>600^2 = 360,000</math></small> distinct chords. With all of its chords ''and their intersections'' it is the 14400 vertex [[W:Omnitruncation|omnitruncated]] 120-cell, which is identical to the omnitruncated 600-cell given the symmetry of their Coxeter-Dynkin diagrams.}} The following animation is an exception which does show some interior chords, although it does not reveal the inscribed 4-polytopes. {| class=wikitable width=540 !colspan=1|Coxeter section views |- |align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]<br>Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls. |} == Related polyhedra and honeycombs== === H₄ polytopes === The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} {{H4_family}} === {p,3,3} polytopes === The 120-cell is similar to three [[W:Regular 4-polytope|regular 4-polytopes]]: the [[5-cell|5-cell]] {3,3,3} and [[W:Tesseract|tesseract]] {4,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[W:Tetrahedral|tetrahedral]] [[W:Vertex figure|vertex figure]] {3,3}: {{Tetrahedral vertex figure tessellations small}} === {5,3,p} polytopes === The 120-cell is a part of a sequence of 4-polytopes and honeycombs with [[W:Dodecahedral|dodecahedral]] cells: {{Dodecahedral_tessellations_small}} === Tetrahedrally diminished 120-cell === Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.{{Efn|The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the [[600-cell#Diminished 600-cells|diminishment of the 120-point 600-cell]] by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point [[W:Snub 24-cell|snub 24-cell]]. Similarly, the 8-cell tesseract can be seen as a 16-point [[24-cell#Diminishings|diminished 24-cell]] from which one 8-point 16-cell has been removed.}} [[File:Tetrahedrally_diminished_regular_dodecahedron.png|thumb|In the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]], 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.]] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]] because the 4 vertices removed formed a [[#Dual 600-cells|tetrahedron inscribed in the dodecahedron]]. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.{{Efn|name=face pentagon chord}} The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron,{{Efn|Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular [[5-cell|5-cell]] with a regular tetrahedron base.|name=truncated apex}} each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges. The '''480-point diminished 120-cell''' may be called the '''tetrahedrally diminished 120-cell''' because its cells are tetrahedrally diminished, or the '''600-cell diminished 120-cell''' because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the '''regular 5-cells diminished 120-cell''' because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.{{Efn|name=inscribed 5-cells}} === Davis 120-cell manifold === The '''Davis 120-cell manifold''', introduced by {{harvtxt|Davis|1985}}, is a compact 4-dimensional [[W:Hyperbolic manifold|hyperbolic manifold]] obtained by identifying opposite faces of the 120-cell, whose universal cover gives the [[W:List of regular polytopes#Tessellations of hyperbolic 4-space|regular honeycomb]] [[W:order-5 120-cell honeycomb|{5,3,3,5}]] of 4-dimensional hyperbolic space. ==See also== *[[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] *[[W:57-cell|57-cell]] – an abstract regular 4-polytope constructed from 57 [[W:Hemi-dodecahedron|hemi-dodecahedra]]. *[[600-cell]] - the dual [[W:4-polytope|4-polytope]] to the 120-cell ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} * {{Citation | last1=Davis | first1=Michael W. | title=A hyperbolic 4-manifold | doi=10.2307/2044771 | year=1985 | journal=[[W:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=93 | issue=2 | pages=325–328| jstor=2044771 }} *[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} * {{Cite journal|last1=Schleimer|first1=Saul|last2=Segerman|first2=Henry|date=2013|title=Puzzling the 120-cell|journal=Notices Amer. Math. Soc.|volume=62|issue=11|pages=1309–1316|doi=10.1090/noti1297 |arxiv=1310.3549 |s2cid=117636740|ref={{SfnRef|Schleimer & Segerman|2013}}}} {{Refend}} ==External links== * [https://www.youtube.com/watch?v=MFXRRW9goTs/ YouTube animation of the construction of the 120-cell] Gian Marco Todesco. * [http://www.theory.org/geotopo/120-cell/ Construction of the Hyper-Dodecahedron] * [http://www.gravitation3d.com/120cell/ 120-cell explorer] &ndash; A free interactive program (requires Microsoft .Net framework) that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL. [[Category:Geometry]] [[Category:Polyscheme]] 71zmcqy6hv8teh0nnd5pood24prq9j7 2806689 2806688 2026-04-26T18:38:57Z Dc.samizdat 2856930 /* Concentric hulls */ 2806689 wikitext text/x-wiki {{Short description|Four-dimensional analog of the dodecahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=120-cell | Image_File=Schlegel wireframe 120-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Snub 24-cell|31]] | Index=32 | Next=[[W:Rectified 120-cell|33]] | Schläfli={5,3,3}| CD={{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}| Cell_List=120 [[W:Dodecahedron|{5,3}]] [[Image:Dodecahedron.png|20px]]| Face_List=720 [[W:Pentagon|{5}]] [[File:Regular pentagon.svg|20px]]| Edge_Count=1200| Vertex_Count= 600| Petrie_Polygon=[[W:Triacontagon|30-gon]]| Coxeter_Group=H₄, [3,3,5]| Vertex_Figure=[[File:120-cell verf.svg|80px]]<br>[[W:Tetrahedron|tetrahedron]]| Dual=[[600-cell]]| Property_List=[[W:Convex set|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} {{maths}} [[File:120-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:Geometry|geometry]], the '''120-cell''' is the [[W:Convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {5,3,3}. It is also called a '''C₁₂₀''', '''dodecaplex''' (short for "dodecahedral complex"), '''hyperdodecahedron''', '''polydodecahedron''', '''hecatonicosachoron''', '''dodecacontachoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hecatonicosahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> The boundary of the 120-cell is composed of 120 dodecahedral [[W:Cell (mathematics)|cells]] with 4 meeting at each vertex. Together they form 720 [[W:Pentagon|pentagonal]] faces, 1200 edges, and 600 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:Regular dodecahedron|regular dodecahedron]], since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the [[600-cell]]. == Geometry == The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|name=4-polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the [[5-cell|5-cell]],{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=''Remark 5.7''|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional [[W:Swiss Army knife|Swiss Army knife]]: it contains one of everything. It is daunting but instructive to study the 120-cell, because it contains examples of ''every'' relationship among ''all'' the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why [[W:John Stillwell|Stillwell]] titled his paper on the 4-polytopes and the history of mathematics<ref>''Mathematics and Its History'', John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}}</ref> of more than 3 dimensions ''The Story of the 120-cell''.{{Sfn|Stillwell|2001}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} ===Cartesian coordinates=== Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen. ==== √8 radius coordinates ==== The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable |- !24 | ({0, 0, ±2, ±2}) | [[24-cell#Great squares|24-cell]] | rowspan=7 | 600-point 120-cell |- !64 | ({±φ, ±φ, ±φ, ±φ⁻²}) | |- !64 | ({±1, ±1, ±1, ±{{radic|5}}<nowiki />}) | |- !64 | ({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²}) | |- !96 | ([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !96 | ([0, ±φ⁻², ±1, ±φ²]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !192 | ([±φ⁻¹, ±1, ±φ, ±2]) | |} where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[W:Golden ratio|golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618. ==== Unit radius coordinates ==== The unit-radius 120-cell has edge length {{Sfrac|1|φ²{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone, Pileio & Levitt|2010|p=1442|loc=Table 3}} are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} in the left column below: {| class="wikitable" style=width:720px |- !rowspan=3|120 !8 |style="white-space: nowrap;"|({±1, 0, 0, 0}) |[[16-cell#Coordinates|16-cell]] | rowspan="2" |[[24-cell#Great hexagons|24-cell]] | rowspan="3" |[[600-cell#Coordinates|600-cell]] | rowspan="10" style="white-space: nowrap;"|120-cell |- !16 |style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2 |[[W:Tesseract#Radial equilateral symmetry|Tesseract]] |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±1, ±φ]) / 2 |colspan=2|[[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !rowspan=7|480 !colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]] !5-point [[5-cell#Coordinates|5-cell]] ![[24-cell#Great squares|24-cell]] ![[600-cell#Coordinates|600-cell]] |- !32 |style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ⁻²]) / {{radic|8}} |rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br> (−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br> (−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4 |rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0}) |rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br> ({±1, ±1, ±1, ±1}) / 2<br> ([0, ±φ⁻¹, ±1, ±φ]) / 2 |- !32 |style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}} |- !32 |style="white-space: nowrap;"|([±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻², ±1, ±φ²]) / {{radic|8}} |- !192 |style="white-space: nowrap;"|([±φ⁻¹, ±1, ±φ, ±2]) / {{radic|8}} |- |colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[W:Quaternion|quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[W:Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone, Pileio & Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone, Pileio & Levitt|2010|loc=Table 3|p=1442}}}} |} The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}} ===Chords=== [[File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also [[W:Clifford parallel|Clifford parallel]]) and they intersect in a single point. In [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotations]], points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a ''helical'' smooth curve called an ''isocline''{{Efn|An '''isocline''' is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a ''straight line'', a [[W:Geodesic|geodesic]]. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are [[W:Link (knot theory)|linked]] and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} To avoid confusion, we always refer to an ''isocline'' as such, and reserve the term ''[[W:Great circle|great circle]]'' for an ordinary great circle in the plane.|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a [[W:SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has 5-cell edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains an irregular great dodecagon compound of several of these great circle polygons in the same central plane, [[#Compound of five 600-cells|illustrated below]].]] {{see also|600-cell#Golden chords}} The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|[[File:Regular_star_figure_6(5,2).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> six of the 120 disjoint regular 5-cells of edge-length {{radic|2.5}} which are inscribed in the 120-cell appear as six pentagrams, the [[5-cell#Boerdijk–Coxeter helix|Clifford polygon of the 5-cell]]. The 30 vertices comprise a Petrie polygon of the 120-cell,{{Efn|name=two coaxial Petrie 30-gons}} with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In [[600-cell#Decagons and pentadecagrams|600-cell § Decagons and pentadecagrams]], see the illustration of [[W:Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, [[5-cell#Geodesics and rotations|pentagonal circuits]] each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two ''disjoint 5-cells'' are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic [[W:SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>2 disjoint [[W:Pentadecagram|pentadecagram]] isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the ''same'' discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}}]]The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and [[5-cell#Geodesics and rotations|its inscribed regular 5-cells' opposing 1200 edges]].{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an [[#Compound of five 600-cells|irregular great dodecagon]], a compound [[#Chords|great circle polygon of the 120-cell]].|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete [[W:Hopf fibration|Hopf fibration]].{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentadecagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=isocline}} of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} and they are occupied by adjacent [[W:Clifford parallel|Clifford parallel]] great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' rotations.{{Efn|name=distinct rotations}}}} They also give the 120-cell a characteristic great circle polygon: an ''irregular'' great hexagon in which three 120-cell edges alternate with three 5-cell edges.{{Efn|name=irregular great hexagon}} The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the [[5-cell#Geodesics and rotations|5-cell]] and the [[W:8-cell|8-cell tesseract]], they form zig-zag [[W:Petrie polygon|Petrie polygon]]s instead.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical ''isocline''{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its ''Clifford polygon''.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[W:Triacontagon|triacontagon]] {30} zig-zag [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[W:Skew polygon|skew]] regular [[W:Triacontagon|triacontagon]] {30}.{{Efn|name=15 distinct chord lengths}} The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel [[#Compound of five 600-cells|{12} great circle polygons]].]] The 120-cell contains 80 distinct [[W:30-gon|30-gon]] Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons.{{Efn|name=Petrie polygons of the 120-cell}} The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.{{Efn|name=two coaxial Petrie 30-gons}} The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has at least 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the [[W:Petrie polygon|Petrie polygon]] can be picked out as a [[W:Skew polygon|skew polygon]] of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix ring]]. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered [[#Chords|chords]] of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct [[W:Pythagorean distance|Pythagorean distance]]s through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord is the {30/''n''} polygram chord, which connects two vertices that are ''n'' edge lengths apart on a Petrie polygon of the 120-cell. The 15 major chords lie on great circles in central planes that contain regular and irregular polygons of {4}, {10}, or {12} vertices. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). The 15 ''minor chords'' lie on rectangular {4} great circles and do not occur anywhere except inside the 120-cell. In this article, we refer to the 15 minor chords by reference to their arc-angles, e.g. 41.4~° #3⁺ with length {{radic|0.5}} falls between the #3 and #4 chords.|name=additional 120-cell chords}} The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[W:Snub 24-cell|snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=4-polytopes ordered by size and complexity}} The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Great hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[W:Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Great hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[W:H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[W:Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column ''a'' for (30−''i'')₀) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}} |- !colspan=6|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}} ![[5-cell|5-cell]] ![[16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[24-cell|24-cell]] ![[W:Snub 24-cell|Snub]] ![[600-cell]] ![[#Tetrahedrally diminished 120-cell|Dimin]] ! style="border-right: none;"|120-cell ! style="border-left: none;"| |- !colspan=6|Vertices | style="background: seashell;"|5 | style="background: paleturquoise;"|8 | style="background: paleturquoise;"|16 | style="background: paleturquoise;"|24 | style="background: yellow;"|96 | style="background: yellow;"|120 | style="background: seashell;"|480 | style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}} |rowspan=6 style="background: seashell; border: none;"| |- !colspan=6|Edges | style="background: seashell;"|10{{Efn|name=irregular great hexagon}} | style="background: paleturquoise;"|24 | style="background: paleturquoise;"|32 | style="background: paleturquoise;"|96 | style="background: yellow;"|432 | style="background: yellow;"|720 | style="background: seashell;"|1200 | style="background: seashell;"|1200{{Efn|name=irregular great hexagon}} |- !colspan=6|Edge chord | style="background: seashell;{{text color default}};"|#8{{Efn|name=inscribed 5-cells}} | style="background: paleturquoise;"|#7 | style="background: paleturquoise;"|#5 | style="background: paleturquoise;"|#5 | style="background: yellow;"|#3 | style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the [[600-cell]] and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-cell#Boerdijk–Coxeter helix rings|600-cell Petrie polygon is a helical ring]] which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a [[W:Triacontagon#Triacontagram|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a [[600-cell#Decagons|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[600-cell#Boerdijk–Coxeter helix rings|600-cell's Petrie helix]] does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew [[W:Triacontagon#Triacontagram|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[W:Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}} | style="background: seashell;"|#1 | style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}} |- !colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[W:SO(4)#Double rotations|double rotation]] in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its ''rotation angle'' and its ''isocline angle''.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this ''incorrectly'', because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its ''completely'' orthogonal plane, but Clifford parallel to it. It rotates ''with'' its completely orthogonal plane, but not ''in'' it. It is Clifford parallel to its completely orthogonal plane ''and'' to the plane it is moving to, and does not intersect them; the plane that it rotates ''in'' is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, [[16-cell#Rotations|all 6 orthogonal planes]] rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different [[W:Orientation entanglement|orientation]] (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} | style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]] | style="background: paleturquoise;"|[[16-cell#Helical construction|#15]] | style="background: paleturquoise;"|#10 | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]] | style="background: yellow;"|#5 | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]] | style="background: seashell;"|#4 | style="background: seashell;"|#4{{Efn|The characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}} |- !colspan=6|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Efn|name=isoclinic}}|name=Clifford polygon}} | style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]] | style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]] | style="background: paleturquoise;"| | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]] | style="background: yellow;"| | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]] | style="background: seashell;"| | style="background: seashell;"|[[W:Pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}} |- !colspan=3|Chord !Arc !colspan=2|Edge | style="background: seashell;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: yellow;"| | style="background: yellow;"| | style="background: seashell;"| | style="background: seashell;"| |- style="background: seashell;"| |rowspan=2|#1<br>△ |rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]] |rowspan=2|30 |{{Efn|name=120-cell Petrie {30}-gon}} |colspan=2|120-cell edge <big>𝛇</big> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |15.5~° |{{radic|𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <small>{{sfrac|1|φ}}</small> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² = <small>{{sfrac|1|φ²}}</small> ≈ 0.382 {{indent|7}}𝜀 = 𝚫²/2 = <small>{{sfrac|1|2φ⁴}}</small> ≈ 0.073<br> and the 120-cell edge-length is: {{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270<br> For example: {{indent|7}}𝛇 = {{radic|𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}} |0.270~ |- style="background: seashell;"| |rowspan=2|#2<br><big>☐</big> |rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]] |rowspan=2|15 | |colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1₀. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[W:Pentagon#Regular pentagons|pentagon diagonal]] (the #2 chord) is in the [[W:Golden ratio|golden ratio]] φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn|name=dodecahedral cell metrics}}|name=face pentagon chord}}|name=#2 chord}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |25.2~° |{{radic|0.19~}} |0.437~ |- style="background: yellow;"| |rowspan=2|#3<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]] |rowspan=2|10 |𝝅/5 |colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |- style="background: seashell;"| |rowspan=2|#4<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]] |rowspan=2|{{sfrac|15|2}} | |colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |44.5~° |{{radic|0.57~}} |0.757~ |- style="background: paleturquoise;"| |rowspan=2|#5<br>△ |rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]] |rowspan=2|6 |𝝅/3 |colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[W:Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a [[W:0-gon|0-gon]] great circle axis that does not intersect any vertices.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different [[#Compound of five 600-cells|central planes]].|name=great hexagon}} |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2|{{green|<big>'''225'''</big>}}<br><br> |rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |60° |{{radic|1}} |1 |- style="background: yellow;"| |rowspan=2|#6<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]] |rowspan=2|5 |2𝝅/5 |colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |- style="background: paleturquoise;"| |rowspan=2|#7<br><big>☐</big> |rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]] |rowspan=2|{{sfrac|30|7}} |𝝅/2 |colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}} |rowspan=2| |rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24 |rowspan=2|{{green|<big>'''675'''</big>}}<br>48 |rowspan=2|<br>72 |rowspan=2| |rowspan=2|<br>1800 |rowspan=2|<br> |rowspan=2|<br>9000 |rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4} |- style="background: paleturquoise;"| |90° |{{radic|2}} |1.414~ |- style="background: #FFCCCC;"| |rowspan=2|#8<br><big>✩</big> |rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]] |rowspan=2|{{sfrac|15|4}} | |colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[W:Triacontagon|triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}} |rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10 |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: #FFCCCC;"| |104.5~° |{{radic|2.5}} |1.581~ |- style="background: yellow;"| |rowspan=2|#9<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]] |rowspan=2|{{sfrac|10|3}} |3𝝅/5 |colspan=2|[[W:Golden section|golden section]] <math>\phi</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |108° |{{radic|2.𝚽}} |1.618~ |- style="background: paleturquoise;"| |rowspan=2|#10<br>△ |rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]] |rowspan=2|3 |2𝝅/3 |colspan=2|[[24-cell#Great triangles|great triangle]] |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2| |rowspan=2|<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |120° |{{radic|3}} |1.732~ |- style="background: seashell;"| |rowspan=2|#11<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]] |rowspan=2|{{sfrac|30|11}} | |colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: yellow;"| |rowspan=2|#12<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]] |rowspan=2|{{sfrac|5|2}} |4𝝅/5 |colspan=2|great [[W:Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: seashell;"| |rowspan=2|#13<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]] |rowspan=2|{{sfrac|30|13}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/13}-gram]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: seashell;"| |rowspan=2|#14<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]] |rowspan=2|{{sfrac|15|7}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/14}=2{15/7}]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200<br> |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: paleturquoise;"| |rowspan=2|#15<br><small>△☐𝜙</small> |rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]] |rowspan=2|2 |𝝅 |colspan=2|[[W:Diameter|diameter]] |rowspan=2| |rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4 |rowspan=2|<br>8 |rowspan=2|<br>12 |rowspan=2|<br>48 |rowspan=2|<br>60 |rowspan=2|<br>240 |rowspan=2|<br>300{{Efn|name=rays and bases}} |rowspan=2|{{blue|<big>'''1'''</big>}}<br><br> |- style="background: paleturquoise;"| |180° |{{radic|4}} |2 |- !colspan=6|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} | style="background: seashell;"|25 | style="background: paleturquoise;"|64 | style="background: paleturquoise;"|256 | style="background: paleturquoise;"|576 | style="background: yellow;"| | style="background: yellow;"|14400 | style="background: seashell;"| | style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}} !<big>{{blue|'''300'''}}</big> |} [[File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.{{Efn|Drawing the fan of chords with #1 and #11 at a different origin than all the others is an artistic choice, since all the chords are incident at every vertex. We could just as well have drawn all the chords from the same origin vertex, but this arrangement notices the parallel relationship between #8 and #11.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] The annotated chord table is a complete [[W:Bill of materials|bill of materials]] for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the '''#3''' chord row, the 600-cell's 72 great decagons contain 720 '''#3''' chords in all. The '''{{red|red}}''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of <big>{{red|'''25'''}}</big> disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''{{green|green}}''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains <big>{{green|'''225'''}}</big> distinct 24-cells which share components. The '''{{blue|blue}}''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the '''#3''' chord row, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[W:Vertex figure|vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell. === Relationships among interior polytopes === The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, ''except'' for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which ''does'' occur. Various regular [[W:Skew polygon|skew polygon]]s {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional [[W:Jigsaw puzzle|jigsaw puzzle]] in which all those polytopes are the parts.{{Sfn|Schleimer & Segerman|2013}} Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.{{Sfn|Stillwell|2001}} The regular 1-polytope occurs in only [[#Chords|15 distinct lengths]] in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By [[W:Alexandrov's uniqueness theorem|Alexandrov's uniqueness theorem]], convex polyhedra with shapes distinct from each other also have distinct [[W:Metric spaces|metric spaces]] of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four {{background color|paleturquoise|[[24-cell#Hypercubic chords|hypercubic chords]]}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built solely from them. {{see also|W:24-cell#Relationships among interior polytopes|label 1=24-cell § Relationships among interior polytopes}} An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|[[600-cell#Golden chords|golden chords]]}} are square roots of irrational fractions that are functions of {{radic|5}}. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built solely from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. {{see also|W:600-cell#Icosahedra|label 1=600-cell § Icosahedra}} All 15 major chords, and 15 other distinct chordal distances (the minor chords [[120-cell#Geodesic rectangles|enumerated below]]), occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{background color|#FFCCCC|[[5-cell#Boerdijk–Coxeter helix|regular 5-cells and {{radic|5/2}} chords]].}}{{Efn|Dodecahedra emerge as ''visible'' features in the 120-cell, but they also occur in the 600-cell as ''interior'' polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the ''regular'' 5-cell (the [[W:Simplex|simplex]] regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the [[W:Tetrahedron|3-simplex]] and [[W:5-orthoplex|5-orthoplex]]. An [[W:simplex|<math>n</math>-simplex]] is bounded by <math>n+1</math> vertices and <math>n+1</math> (<math>n</math>-1)-simplex facets, and has <math>z+1</math> long diameters (its edges) of length <math>\sqrt{n+1}/\sqrt{n}</math> radii. An [[W:orthoplex|<math>n</math>-orthoplex]] is bounded by <math>2n</math> vertices and <math>2^n</math> (<math>n</math>-1)-simplex facets, and has <math>n</math> long diameters (its orthogonal axes) of length <math>2</math> radii. An [[W:hypercube|<math>n</math>-cube]] is bounded by <math>2^n</math> vertices and <math>2n</math> (<math>n</math>-1)-cube facets, and has <math>2^{n-1}</math> long diameters of length <math>\sqrt{n}</math> radii.{{Efn|The <math>n</math>-simplex's facets are larger than the <math>n</math>-orthoplex's facets. For <math>n=4</math>, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of <math>\sqrt{5}</math> to <math>\sqrt{4}</math> to <math>\sqrt{2}</math>.|name=root 5/root 4/root 2}} The <math>\sqrt{3}</math> long diameters of the 3-cube are shorter than the <math>\sqrt{4}</math> axes of the 3-orthoplex. The [[16-cell#Coordinates|coordinates of the 4-orthoplex]] are the permutations of <math>(0,0,0,\pm 1)</math>, and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of <math>(0,0,0,1)</math>.{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting <math>(0,0,0,1)</math>, and its completely orthogonal 3-facet permuting <math>(0,0,0,-1)</math>, comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-[[W:demihypercube|demicube]], half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to <math>n=4</math>. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} The <math>\sqrt{4}</math> long diameters of the 4-cube are the same length as the <math>\sqrt{4}</math> axes of the 4-orthoplex. The [[W:5-orthoplex#Cartesian coordinates|coordinates of the 5-orthoplex]] are the permutations of <math>(0,0,0,0,\pm 1)</math>, and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of <math>(0,0,0,0,1)</math>.{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting <math>(0,0,0,0,1)</math>, and its completely orthogonal 4-facet permuting <math>(0,0,0,0,-1)</math>, comprise all 10 vertices of the 5-orthoplex.}} The <math>\sqrt{5}</math> long diameters of the 5-cube are longer than the <math>\sqrt{4}</math> axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells.{{Efn|No vertex pair of any of the 120 5-cells (no [[5-cell#Geodesics and rotations|great digon central plane of a 5-cell]]) occurs in any of the 675 16-cells (the 675 [[16-cell#Coordinates|Cartesian basis sets of 6 orthogonal central planes]]).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}} {{see also|W:5-cell#Geodesics and rotations|label 1=5-cell § Geodesics and rotations}} === Compound of five 600-cells === [[File:Great dodecagon of the 120-cell.png|thumb|300px|The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).]] The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} As a consequence of being a compound of five disjoint 600-cells, the 120-cell has 200 irregular great dodecagon {12} central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 {12} central planes originate as the compounds of the hexagonal central planes of the 25 disjoint inscribed 24-cells and the digon central planes of the 120 disjoint inscribed regular 5-cells; they contain all the 24-cell and 5-cell edges, and also the 120-cell edges. Thus the edges and characteristic rotations{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The '''characteristic isoclinic rotation of a 4-polytope''' is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each [[W:Hopf fibration|Hopf fibration]] of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an [[#Compound of five 600-cells|irregular great dodecagon]] which also has #4 chord edges. In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in these same 200 rotation planes.{{Efn|name=edge rotation planes}} Each of the ten 600-cells occupies the entire set of 200 planes. The 120-cell's irregular [[#Other great circle constructs|dodecagon {12} great circle polygon]] has 6 short edges (#1 [[#Chords|chords]] marked {{Color|red|𝜁}}) alternating with 6 longer dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} Inscribed in the irregular great dodecagon are two irregular great hexagons ({{color|red|red}}) in alternate positions.{{Efn|name=irregular great hexagon}} Two ''regular'' great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of 200 dodecagon planes. They occur as 100 completely orthogonal pairs, and they are the ''same'' 200 central planes each containing a [[600-cell#Hexagons|hexagon]] that are found in ''each'' of the 10 inscribed 600-cells. There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares regular hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint regular hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells. ===Concentric hulls=== [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px| Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an Overall Hull that is a [[W:Chamfered dodecahedron|chamfered dodecahedron]] of Norm={{radic|8}}.<br /> Hulls 1 - 8 are the 8 sections of the 120-cell beginning with a cell (Hull 1).<br /> Hulls 1, 2, & 7 are each pairs of [[W:Dodecahedron|dodecahedron]]s.<br /> Hull 3 is a pair of [[W:Icosidodecahedron|icosidodecahedron]]s.<br /> Hulls 4 & 5 are each pairs of [[W:Truncated icosahedron|truncated icosahedron]]s.<br /> Hull 6 is a pair of semi-regular [[W:Rhombicosidodecahedron|rhombicosidodecahedron]]s.<br /> Hull 8 is a single non-uniform [[W:Rhombicosidodecahedron#Names|rhombicosidodecahedron]], the central section.<br /> A more detailed visualization of these 15 simplified sections, with subgroup sections where the inscribed solid has more than one permutation in its orbit, is available [https://commons.wikimedia.org/wiki/File:Cell_First_533_120-Cell_Sections.svg here].]] {{Clear}} These hulls illustrate Coxeter's sections 1₃ - 8₃ of the 120-cell, the sections beginning with a cell (hull #1).{{Sfn|Coxeter|1973|p=299|loc=Table V (iv) Sections of {5,3,3} beginning with a cell (right half of table)}} A ''section'' is a flat 3-dimensional hyperplane slice through the [[W:3-sphere|3-sphere]]: a 2-sphere (ordinary sphere). It is dimensionally analogous to a flat 2-dimensional plane slice through a 2-sphere: a 1-sphere (ordinary circle). The hulls are illustrated as if they were all the same size, but actually they increase in radius as numbered: they are concentric 2-spheres that nest inside each other. Every cell of the 120-cell is the smallest hull in its own set of 8 concentric hulls. There are 120 distinct sets of hulls. The 120-cell actually has 15 sections beginning with a cell, numbered 1 - 15 with number 8 in the center. After increasing in size from 1 to 8, the hulls get smaller again. Sections 1 and 15 are both a hull #1, the smallest hull, a dodecahedral cell of the 120-cell. Section #8 is the central section, the largest hull, with the same radius as the 120-cell. Except for the central section #8, the sections occur in parallel pairs, on either side of the central section. Hull #8 is dimensionally analogous to the equator, while hulls #1 - #7 are dimensionally analogous to lines of latitude. There are 120 of each kind of hull #1 - #7 in the 120-cell, but only 60 of the central hull #8. {{Clear}} The 120-cell also has 30 sections beginning with a vertex, illustrated below. Like the sections beginning with a cell illustrated above, the vertex-first sections are also flat 3-dimensional hyperplane slices through the 3-sphere, polyhedra that nest inside each other as concentric 2-spheres. Section 0₀ is the vertex itself. Section 1₀ is the 120-cell's tetrahedral vertex figure. Sections 1₀ - 29₀ are described in more detail in [[120-cell#Geodesic rectangles|§Geodesic rectangles]] below. {{Clear}} [[File:Vertex_First_533_120-Cell_Sections.svg|thumb|left|640px| Coxeter's sections 0₀ - 30₀ of the 120-cell, the sections beginning with a vertex, showing the orbit sections and subgroup sections (when the inscribed solid has more than one permutation in its orbit), as well as the convex hull of each orbit on the right.]] {{Clear}} === Geodesic rectangles === The 30 distinct chords{{Efn|name=additional 120-cell chords}} found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Background color|palegreen|△ planes that intersect {12} vertices}} in an [[#Compound of five 600-cells|irregular great dodecagon]], {{Background color|yellow|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and <big>☐</big> planes that intersect {4} vertices in several kinds of {{Background color|gainsboro|rectangle}}, including a {{Background color|seashell|square}}. Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing [[#Concentric hulls|polyhedral sections]] of the 120-cell beginning with a vertex (the 0₀ section), as illustrated above. The correspondence is that each 120-cell vertex is surrounded in curved 3-space <math>S_3</math> by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius in Euclidean 3-space <math>R_3</math>.{{Efn|In the curved 3-dimensional space <math>S_3</math> of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the ''apex'' vertex and its surrounding ''base'' polyhedron form a [[W:Polyhedral pyramid|polyhedral pyramid]]. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} There are 600 distinct sets of 15 hulls. The #1 chord is the radius in <math>S_3</math> of the 1₀ section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the radius in <math>S_3</math> of its congruent opposing 29₀ section. The #7 chord is the radius in <math>S_3</math> of the central vertex-first section of the 120-cell, in which two opposing 15₀ sections are coincident. Each vertex is surrounded by two instances of each polyhedron, at the near and far radial distances of the polyhedron's 180° complementary chords, but because curved space <math>S_3</math> begins to close back up on itself after the #7 90° chord, the near and far concentric polyhedra are the same size. Each chord length is given three ways (on successive lines): for the unit-radius 120-cell as a square root, for the unit-radius 120-cell, and for the unit-edge 120-cell.{{Efn|We give chord lengths as unit-radius square roots in these articles, even when they are integers (e.g. the long diameter is {{radic|4}}). Our usual metric is unit-radius, which reveals relationships among successive 4-polytopes,{{Efn|name=4-polytopes ordered by size and complexity}} but Coxeter{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} and Steinbach{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with "fan of chords" diagrams.|p=23|loc=Figure 3}} use unit-edge, which reveals relationships among successive chords.|name=metrics}} To the left of this last unit-edge metric, its reciprocal^-1 is given. The reciprocal is the long radius of a regular ''n''₀-polygon with unit-radius 120-cell edges (#1 chords) as its edges; but this does not imply that the section ''n''₀ polyhedron contains any ''n''₀ polygons.{{Efn|The 120-cell contains no regular {30} central polygons, although its Petrie polygon is a skew regular {30}. Therefore the edge of the regular triacontagon {30} is not a chord of the 120-cell represented in this table. Nevertheless these metrics of the {30} are relevant:<br> :Unit-radius {30}: ::Edge <small><math>E = 2 \sin{\pi/30} \approx \sqrt{0.0437} \approx 0.209</math></small> :Unit-edge {30}: ::Radius <small><math>R_{ue} = 1/E \approx 4.783</math></small> :{30} with 120-cell edges: ::Edge <small><math>\zeta \approx 0.270~</math></small> ::<small><math>E \approx 0.774 \times \zeta</math></small> ::Radius <small><math>R_\zeta \approx 1.292</math></small> |name=triacontagon metrics}} {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |#0<br><br>0₀ | |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |<math>\pi</math> |{{radic|4}} |{{radic|4}} | rowspan="3" |#15<br><br>30₀ |- style="background: palegreen;" | |0° |0 |0 |180° |2 |2 |- style="background: palegreen;" | | |0 |<small><math>0\times\zeta</math></small> |0.135~^-1 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#1<br><br>1₀ |𝞯 |{{radic|0.𝜀}}{{Efn|name=fractional square roots}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} | |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |#14<br><br>29₀ |- style="background: palegreen;" | |15.5~°{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}} |0.270~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |164.5~° |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |1^-1 |1 |<small><math>1\times\zeta</math></small> |0.136~^-1 |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#2<br><br>2₀ |{{Efn|name=#2 chord}} |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 | |{{radic|3.81~}} | | rowspan="3" |#13<br><br>28₀ |- style="background: gainsboro;" | |25.2~° |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |154.8~° |1.952~ | |- style="background: gainsboro;" | |0.618~^-1 |1.618~ |<small><math>\phi\times\zeta</math></small> |0.138~^-1 |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |#3<br><br>3₀ |<math>\pi / 5</math> |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |<math>4\pi / 5</math> |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |#12<br><br>27₀ |- style="background: yellow;" | |36° |0.618~ |<small><math>1 / \phi</math></small> |144°{{Efn|name=dihedral}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.437~^-1 |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |0.142~^-1 |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#3⁺<br><br>4₀ | |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |#12⁻<br><br>26₀ |- style="background: gainsboro;" | |41.4~° |0.707~ |<small><math>\sqrt{2}/2</math></small> |138.6~° |1.871~ | |- style="background: gainsboro;" | |0.382~^-1 |2.618~ |<small><math>\phi^2\times\zeta</math></small> |0.144~^-1 |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#4<br><br>5₀ | |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown). Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' great hexagons (the [[600-cell#Hexagons|24-cell's characteristic rotation]]), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in ''irregular'' great hexagons).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every ''regular'' great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.|name=#4 isocline chord bridge}}|name=dodecagon rotation}}<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" |{{Efn|name=#4 isocline chord bridge}} | |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |#11<br><br>25₀ |- style="background: palegreen;" | |44.5~° |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |135.5~° |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.357~^-1 |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |0.146~^-1 |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#4⁺<br><br>6₀ | |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |#11⁻<br><br>24₀ |- style="background: gainsboro;" | |49.1~° |0.831~ | |130.9~° |1.819~ | |- style="background: gainsboro;" | |0.325~^-1 |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |0.148~^-1 |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁻<br><br>7₀ | |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |#10⁺<br><br>23₀ |- style="background: gainsboro;" | |56° |0.939~ | |124° |1.766~ | |- style="background: gainsboro;" | |0.288~^-1 |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |0.153~^-1 |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |#5<br><br>8₀ |<math>\pi / 3</math> |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}}<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |<small><math>2\pi / 3</math></small> |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |#10<br><br>22₀ |- style="background: palegreen;" | |60° |1 | |120° |1.732~ | |- style="background: palegreen;" | |0.270~^-1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |0.156~^-1 |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁺<br><br>9₀ | |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |#10⁻<br><br>21₀ |- style="background: gainsboro;" | |66.1~° |1.091~ | |113.9~° |1.676~ | |- style="background: gainsboro;" | |0.247~^-1 |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |0.161~^-1 |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁻<br><br>10₀ | |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |#9⁺<br><br>20₀ |- style="background: gainsboro;" | |69.8~° |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |110.2~° |1.640~ | |- style="background: gainsboro;" | |0.236~^-1 |4.236~ |<small><math>\phi^3\times\zeta</math></small> |0.165~^-1 |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |#6<br><br>11₀ |<math>2\pi/5</math> |{{radic|1.𝚫}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}}<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |<math>3\pi / 5</math> |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |#9<br><br>19₀ |- style="background: yellow;" | |72° |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |108° |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |0.230~^-1 |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |0.167~^-1 |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |#6⁺⁻<br><br>12₀ | |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic|2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12₀ sections and 18₀ sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}}<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 | |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |#8<br><br>18₀ |- style="background: palegreen;" | |75.5~° |1.224~ | |104.5~° |1.581~ | |- style="background: palegreen;" | |0.221~^-1 |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |0.171~^-1 |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺<br><br>13₀ | |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.31~}} | | rowspan="3" |#8⁻<br><br>17₀ |- style="background: gainsboro;" | |81.1~° |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |98.9~° |1.520~ | |- style="background: gainsboro;" | |0.208~⁻¹ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |0.178~^-1 |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺⁺<br><br>14₀ | |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |#7⁺<br><br>16₀ |- style="background: gainsboro;" | |84.5~° |1.345~ | |95.5~° |1.480~ | |- style="background: gainsboro;" | |0.201~⁻¹ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |0.182~^-1 |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |#7<br><br>15₀ |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}}<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |#7<br><br>15₀ |- style="background: seashell;" | |90° |1.414~ | |90° |1.414~ | |- style="background: seashell;" | |0.191~⁻¹ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |0.191~^-1 |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|name=isoclinic rotation}} of a distinct class,{{Efn|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons" / 4.}} In each class of rotation,{{Efn|[[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]] are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are ''invariant'', which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually [[W:Clifford parallel|Clifford parallel]]. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called ''fibrations'', which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular ''fibers'', the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} ===Polyhedral graph=== Considering the [[W:Adjacency matrix|adjacency matrix]] of the vertices representing the polyhedral graph of the unit-radius 120-cell, the [[W:Graph diameter|graph diameter]] is 15, connecting each vertex to its coordinate-negation at a [[W:Euclidean distance|Euclidean distance]] of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are [[W:Vertex coloring|3-colorable]]. The graph is [[W:Eulerian path|Eulerian]] having degree 4 in every vertex. Its edge set can be decomposed into two [[W:Hamiltonian path|Hamiltonian cycles]].<ref>{{cite book| author = Carlo H. Séquin | title = Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes | date = July 2005 | pages = 463–472 | publisher = Mathartfun.com | isbn = 9780966520163 | url = https://archive.bridgesmathart.org/2005/bridges2005-463.html#gsc.tab=0 | access-date=March 13, 2023}}</ref> === Constructions === The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the [[600-cell]],{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the [[24-cell|24-cell]],{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 24-cell}} the 24-cell can be deconstructed into three distinct instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 '''rays''' [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a '''[[W:Orthonormal basis|basis]]'''. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 300₉675₄ to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct [[16-cell#Coordinates|16-cell]] containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse [[W:Golden ratio|golden ratio]]), but the 120-cell's edge length is ~0.270 times its radius. The 120-cell is also the convex hull of the regular compound of 120 disjoint regular 5-cells. This can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point 600-cell, expand each vertex into a regular 5-cell. For each of the 120 vertices, add 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point 600-cells: a 120-cell. ==== Dual 600-cells ==== [[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]] Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ({{sfrac|φ²|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ²{{radic|2}} ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The [[#Unit radius coordinates|unit radius 120-cell]] (with edge-length {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ²}} ≈ 1.080. [[File:Dodecahedron_vertices.svg|thumb|150px|right|One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.{{Efn|In the [[W:120-cell#Dual 600-cells|dodecahedral cell]] of the unit-radius 120-cell, the edge is the '''15.5° #1 [[#Chords|chord]]''' of the 120-cell of length <small><math>\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270</math></small>. Eight {{Color|orange}} vertices lie at the Cartesian coordinates <small><math>(\pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8})</math></small> relative to origin at the cell center. They form a cube (dashed lines) whose edges are the '''25.2° #2 chord''' of length <small><math>\tfrac{1}{\phi\sqrt{2}} \approx 0.437</math></small> (the pentagon diagonal). The face diagonals of the cube (not drawn) are the '''36° #3 chord''' of length <small><math>\tfrac{1}{\phi} \approx 0.618</math></small> (the edges of two 600-cell tetrahedron cells inscribed in the cube). The next largest '''41.1° chord''' has length <small><math>\tfrac{1}{\sqrt{2}} \approx 0.707</math></small>. The diameter of the dodecahedron is the '''44.5° #4 chord''' of length <small><math>\tfrac{\sqrt{3}}{\phi\sqrt{2}} \approx 0.757</math></small> (the cube diagonal). If the #4 diameter is extended outside the dodecahedron in a straight line in the curved space of the 3-sphere, it is colinear with a #1 edge belonging to three neighboring dodecahedron cells, and the combined '''60° #5 chord''' has length <small><math>\sqrt{1}</math></small> (an edge of an inscribed 24-cell). If this 60° combined #4 plus #1 geodesic is further extended in a straight line by another #4 chord (the diameter of a further cell), the combined '''104.5° #8 chord''' has length <small><math>\tfrac{\sqrt{5}}{\sqrt{2}} \approx 1.581</math></small> (an edge of an inscribed regular 5-cell).|name=dodecahedral cell metrics}}]] Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ²|{{radic|8}}}} ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given [[#√8 radius coordinates|above]] of long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ²}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ², which is smaller than {{Radic|8}} in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ²{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps. ==== Cell rotations of inscribed duals ==== Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} This shows that the 120-cell contains, among its many interior features, 120 [[W:Compound of ten tetrahedra|compounds of ten tetrahedra]], each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because [[#Cell rotations of inscribed duals|each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells]]. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 600-cells}} All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-cell.}} Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. ==== Augmentation ==== Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing [[W:Hyperpyramid|4-pyramid]]s of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a [[W:Regular dodecahedron#Cartesian coordinates|dodecahedron]].{{Efn|name=truncated apex}} Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a [[600-cell#Icosahedra|cluster of five tetrahedra]], with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.{{Efn|As we saw in the [[600-cell#Cell clusters|600-cell]], these 12 tetrahedra belong (in pairs) to the 6 [[600-cell#Icosahedra|icosahedral clusters]] of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. ==== Weyl orbits ==== Another construction method uses [[W:Quaternion|quaternion]]s and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe <math>T</math> and <math>T'</math> [[24-cell|24-cell]]s as quaternion orbit weights of D4 under the Weyl group W(D4):<br/> O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}<br/> O(1000) : V1<br/> O(0010) : V2<br/> O(0001) : V3 <math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math> With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the 120-cell of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the [[600-cell]] <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> * the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math> * the [[W:Dual snub 24-cell|dual snub 24-cell]] = <math>T \oplus T' \oplus S'</math>. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}} <math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} ! [[W:K-face|''k''-face]]||f_k||f₀ || f₁||f₂||f₃||[[W:vertex figure|''k''-fig]] !Notes |- align=right |A₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|3|node}} ||( ) !f₀ || 600 || 4 || 6 || 4 ||[[W:Regular tetrahedron|{3,3}]] || H₄/A₃ = 14400/24 = 600 |- align=right |A₁A₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|3|node}} ||{ } !f₁ || 2 || 1200 || 3 || 3 || [[W:Equilateral triangle|{3}]] || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |H₂A₁ ||{{Coxeter–Dynkin diagram|node_1|5|node|2|node_x|2|node}} ||[[W:Pentagon|{5}]] !f₂ || 5 || 5 || 720 || 2 || { } || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |H₃ ||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|2|node_x}} ||[[W:Regular dodecahedron|{5,3}]] !f₃ || 20 || 30 || 12 ||120|| ( ) || H₄/H₃ = 14400/120 = 120 |} == Visualization == The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[24-cell|24-cell]]). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}} === Layered stereographic projection === The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer & Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12). Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an [[W:Icosidodecahedron|icosidodecahedron]], with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="4" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 12 cells | First layer of meridional cells / "[[W:Arctic Circle|Arctic Circle]]" | style="text-align: center" | 36° |- | style="text-align: center" | 3 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 60° |- | style="text-align: center" | 4 | style="text-align: center" | 12 cells | Second layer of meridional cells / "[[W:Tropic of Cancer|Tropic of Cancer]]" | style="text-align: center" | 72° |- | style="text-align: center" | 5 | style="text-align: center" | 30 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" | Equator |- | style="text-align: center" | 6 | style="text-align: center" | 12 cells | Third layer of meridional cells / "[[W:Tropic of Capricorn|Tropic of Capricorn]]" | style="text-align: center" | 108° | rowspan="4" | Southern Hemisphere |- | style="text-align: center" | 7 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 120° |- | style="text-align: center" | 8 | style="text-align: center" | 12 cells | Fourth layer of meridional cells / "[[W:Antarctic Circle|Antarctic Circle]]" | style="text-align: center" | 144° |- | style="text-align: center" | 9 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 120 cells ! colspan="3" | |} The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell. === Intertwining rings === [[Image:120-cell rings.jpg|right|thumb|300px|Two intertwining rings of the 120-cell.]] [[File:120-cell_two_orthogonal_rings.png|thumb|300px|Two orthogonal rings in a cell-centered projection]] The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized [[W:Hopf fibration|Hopf fibration]].{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer & Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical [[W:Torsion of a curve|torsion]]. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two ([[W:Clifford torus|Clifford]]) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B²|pp=23-29}} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint ([[W:Clifford parallel|Clifford parallel]]) great circles. === Other great circle constructs === There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter [[#Chords|chords]], forming an [[#Compound of five 600-cells|irregular dodecagon in a central plane]]. Both these great circle paths have dual [[600-cell#Union of two tori|great circle paths in the 600-cell]]. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a [[600-cell#Decagons|decagon]].{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six [[W:Triangular bipyramids|triangular bipyramids]]) in the 600-cell. This latter path corresponds to a [[600-cell#Icosahedra|ring of six icosahedra]] meeting face to face in the [[W:Snub 24-cell|snub 24-cell]] (or [[W:Icosahedral pyramid|icosahedral pyramids]] in the 600-cell), forming a [[600-cell#Hexagons|hexagon]]. Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges [[#Chords|illustrated above]].{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each [[#Compound of five 600-cells|irregular great dodecagon]], in alternate positions. === 2D Orthogonal projections === [[W:Orthographic projection|Orthogonal projection]]s of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by [[W:B. L. Chilton|B. L. Chilton]].{{Sfn|Chilton|1964}} The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:Van Oss polygon|van Oss polygon]]. {| class="wikitable" |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:120-cell graph H4.svg|240px]]<br>[30]<br>(Red=1) |[[File:120-cell t0 p20.svg|240px]]<br>[20]<br>(Red=1) |[[File:120-cell t0 F4.svg|240px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:120-cell t0 H3.svg|240px]]<br>[10]<br>(Red=5, orange=10) |[[File:120-cell t0 A2.svg|240px]]<br>[6]<br>(Red=1, orange=3, yellow=6, lime=9, green=12) |[[File:120-cell t0 A3.svg|240px]]<br>[4]<br>(Red=1, orange=2, yellow=4, lime=6, green=8) |} === 3D Perspective projections === These projections use [[W:Perspective projection|perspective projection]], from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. A comparison of perspective projections of the 3D dodecahedron to 2D (above left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. [[W:Schlegel diagram|Schlegel diagram]]s use [[W:Perspective (graphical)|perspective]] to show depth in the dimension which has been flattened, choosing a view point ''above'' a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. [[W:Stereographic projection|Stereographic projection]]s use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a [[W:3-sphere|3-sphere]]. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating [[120-cell#Animations|animations]] below, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must). {| class="wikitable" style="width:540px;" |+Comparison with regular dodecahedron |- !width=80|Projection ![[W:Dodecahedron|Dodecahedron]] !120-cell |- ![[W:Schlegel diagram|Schlegel diagram]] |align=center|[[Image:Dodecahedron schlegel.svg|220px]]<br>12 pentagon faces in the plane |align=center|[[File:Schlegel wireframe 120-cell.png|220px]]<br>120 dodecahedral cells in 3-space |- ![[W:Stereographic projection|Stereographic projection]] |align=center|[[Image:Dodecahedron stereographic projection.png|220px]] |align=center|[[Image:Stereographic polytope 120cell faces.png|220px]]<br>With transparent faces |} {|class="wikitable" |- !colspan=2|Enhanced perspective projections |- |align=center|[[Image:120-cell perspective-cell-first-02.png|240px]] |Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: * Nearest dodecahedron to the 4D viewpoint rendered in yellow * The 12 dodecahedra immediately adjoining it rendered in cyan; * The remaining dodecahedra rendered in green; * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image. |- |align=center|[[Image:120-cell perspective-vertex-first-02.png|240px]] |Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: * Four cells surrounding nearest vertex shown in 4 colors * Nearest vertex shown in white (center of image where 4 cells meet) * Remaining cells shown in transparent green * Cells facing away from 4D viewpoint culled for clarity |} === Animations === {|class="wikitable" !colspan=2|Projections to 3D of a 4D 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]] |- |align=center|[[File:120-cell.gif|256px]] |align=center|[[File:120-cell-inner.gif|256px]] |- |From outside the [[W:3-sphere|3-sphere]] in 4-space. |Inside the [[600-cell#Boundary envelopes|3D surface]] of the 3-sphere. |} In all the above projections of the 120-cell, only the edges of the 120-cell appear. All the other [[#Chords|chords]] are not shown. 600 chords converge at ''each'' of the 600 vertices. The complex [[#Relationships among interior polytopes|interior parts]] of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.{{Efn|[[File:Omnitruncated_120-cell_Coxeter_sections-subsections_projected_from_4D.svg|thumb|A full display of each section's orbits along with sub-section orbits in the 14400-point omnitruncated 120-cell.]]The 120-cell has <small><math>600^2 = 360,000</math></small> distinct chords. With all of its chords ''and their intersections'' it is the 14400 vertex [[W:Omnitruncation|omnitruncated]] 120-cell, which is identical to the omnitruncated 600-cell given the symmetry of their Coxeter-Dynkin diagrams.}} The following animation is an exception which does show some interior chords, although it does not reveal the inscribed 4-polytopes. {| class=wikitable width=540 !colspan=1|Coxeter section views |- |align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]<br>Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls. |} == Related polyhedra and honeycombs== === H₄ polytopes === The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} {{H4_family}} === {p,3,3} polytopes === The 120-cell is similar to three [[W:Regular 4-polytope|regular 4-polytopes]]: the [[5-cell|5-cell]] {3,3,3} and [[W:Tesseract|tesseract]] {4,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[W:Tetrahedral|tetrahedral]] [[W:Vertex figure|vertex figure]] {3,3}: {{Tetrahedral vertex figure tessellations small}} === {5,3,p} polytopes === The 120-cell is a part of a sequence of 4-polytopes and honeycombs with [[W:Dodecahedral|dodecahedral]] cells: {{Dodecahedral_tessellations_small}} === Tetrahedrally diminished 120-cell === Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.{{Efn|The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the [[600-cell#Diminished 600-cells|diminishment of the 120-point 600-cell]] by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point [[W:Snub 24-cell|snub 24-cell]]. Similarly, the 8-cell tesseract can be seen as a 16-point [[24-cell#Diminishings|diminished 24-cell]] from which one 8-point 16-cell has been removed.}} [[File:Tetrahedrally_diminished_regular_dodecahedron.png|thumb|In the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]], 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.]] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]] because the 4 vertices removed formed a [[#Dual 600-cells|tetrahedron inscribed in the dodecahedron]]. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.{{Efn|name=face pentagon chord}} The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron,{{Efn|Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular [[5-cell|5-cell]] with a regular tetrahedron base.|name=truncated apex}} each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges. The '''480-point diminished 120-cell''' may be called the '''tetrahedrally diminished 120-cell''' because its cells are tetrahedrally diminished, or the '''600-cell diminished 120-cell''' because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the '''regular 5-cells diminished 120-cell''' because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.{{Efn|name=inscribed 5-cells}} === Davis 120-cell manifold === The '''Davis 120-cell manifold''', introduced by {{harvtxt|Davis|1985}}, is a compact 4-dimensional [[W:Hyperbolic manifold|hyperbolic manifold]] obtained by identifying opposite faces of the 120-cell, whose universal cover gives the [[W:List of regular polytopes#Tessellations of hyperbolic 4-space|regular honeycomb]] [[W:order-5 120-cell honeycomb|{5,3,3,5}]] of 4-dimensional hyperbolic space. ==See also== *[[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] *[[W:57-cell|57-cell]] – an abstract regular 4-polytope constructed from 57 [[W:Hemi-dodecahedron|hemi-dodecahedra]]. *[[600-cell]] - the dual [[W:4-polytope|4-polytope]] to the 120-cell ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} * {{Citation | last1=Davis | first1=Michael W. | title=A hyperbolic 4-manifold | doi=10.2307/2044771 | year=1985 | journal=[[W:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=93 | issue=2 | pages=325–328| jstor=2044771 }} *[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} * {{Cite journal|last1=Schleimer|first1=Saul|last2=Segerman|first2=Henry|date=2013|title=Puzzling the 120-cell|journal=Notices Amer. Math. Soc.|volume=62|issue=11|pages=1309–1316|doi=10.1090/noti1297 |arxiv=1310.3549 |s2cid=117636740|ref={{SfnRef|Schleimer & Segerman|2013}}}} {{Refend}} ==External links== * [https://www.youtube.com/watch?v=MFXRRW9goTs/ YouTube animation of the construction of the 120-cell] Gian Marco Todesco. * [http://www.theory.org/geotopo/120-cell/ Construction of the Hyper-Dodecahedron] * [http://www.gravitation3d.com/120cell/ 120-cell explorer] &ndash; A free interactive program (requires Microsoft .Net framework) that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL. [[Category:Geometry]] [[Category:Polyscheme]] 12n35b08giivs1o6g1ojc4yie7chmij 2806690 2806689 2026-04-26T18:40:28Z Dc.samizdat 2856930 /* Concentric hulls */ 2806690 wikitext text/x-wiki {{Short description|Four-dimensional analog of the dodecahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=120-cell | Image_File=Schlegel wireframe 120-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Snub 24-cell|31]] | Index=32 | Next=[[W:Rectified 120-cell|33]] | Schläfli={5,3,3}| CD={{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}| Cell_List=120 [[W:Dodecahedron|{5,3}]] [[Image:Dodecahedron.png|20px]]| Face_List=720 [[W:Pentagon|{5}]] [[File:Regular pentagon.svg|20px]]| Edge_Count=1200| Vertex_Count= 600| Petrie_Polygon=[[W:Triacontagon|30-gon]]| Coxeter_Group=H₄, [3,3,5]| Vertex_Figure=[[File:120-cell verf.svg|80px]]<br>[[W:Tetrahedron|tetrahedron]]| Dual=[[600-cell]]| Property_List=[[W:Convex set|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} {{maths}} [[File:120-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:Geometry|geometry]], the '''120-cell''' is the [[W:Convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {5,3,3}. It is also called a '''C₁₂₀''', '''dodecaplex''' (short for "dodecahedral complex"), '''hyperdodecahedron''', '''polydodecahedron''', '''hecatonicosachoron''', '''dodecacontachoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hecatonicosahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> The boundary of the 120-cell is composed of 120 dodecahedral [[W:Cell (mathematics)|cells]] with 4 meeting at each vertex. Together they form 720 [[W:Pentagon|pentagonal]] faces, 1200 edges, and 600 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:Regular dodecahedron|regular dodecahedron]], since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the [[600-cell]]. == Geometry == The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|name=4-polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the [[5-cell|5-cell]],{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=''Remark 5.7''|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional [[W:Swiss Army knife|Swiss Army knife]]: it contains one of everything. It is daunting but instructive to study the 120-cell, because it contains examples of ''every'' relationship among ''all'' the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why [[W:John Stillwell|Stillwell]] titled his paper on the 4-polytopes and the history of mathematics<ref>''Mathematics and Its History'', John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}}</ref> of more than 3 dimensions ''The Story of the 120-cell''.{{Sfn|Stillwell|2001}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} ===Cartesian coordinates=== Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen. ==== √8 radius coordinates ==== The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable |- !24 | ({0, 0, ±2, ±2}) | [[24-cell#Great squares|24-cell]] | rowspan=7 | 600-point 120-cell |- !64 | ({±φ, ±φ, ±φ, ±φ⁻²}) | |- !64 | ({±1, ±1, ±1, ±{{radic|5}}<nowiki />}) | |- !64 | ({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²}) | |- !96 | ([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !96 | ([0, ±φ⁻², ±1, ±φ²]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !192 | ([±φ⁻¹, ±1, ±φ, ±2]) | |} where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[W:Golden ratio|golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618. ==== Unit radius coordinates ==== The unit-radius 120-cell has edge length {{Sfrac|1|φ²{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone, Pileio & Levitt|2010|p=1442|loc=Table 3}} are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} in the left column below: {| class="wikitable" style=width:720px |- !rowspan=3|120 !8 |style="white-space: nowrap;"|({±1, 0, 0, 0}) |[[16-cell#Coordinates|16-cell]] | rowspan="2" |[[24-cell#Great hexagons|24-cell]] | rowspan="3" |[[600-cell#Coordinates|600-cell]] | rowspan="10" style="white-space: nowrap;"|120-cell |- !16 |style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2 |[[W:Tesseract#Radial equilateral symmetry|Tesseract]] |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±1, ±φ]) / 2 |colspan=2|[[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !rowspan=7|480 !colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]] !5-point [[5-cell#Coordinates|5-cell]] ![[24-cell#Great squares|24-cell]] ![[600-cell#Coordinates|600-cell]] |- !32 |style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ⁻²]) / {{radic|8}} |rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br> (−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br> (−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4 |rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0}) |rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br> ({±1, ±1, ±1, ±1}) / 2<br> ([0, ±φ⁻¹, ±1, ±φ]) / 2 |- !32 |style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}} |- !32 |style="white-space: nowrap;"|([±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻², ±1, ±φ²]) / {{radic|8}} |- !192 |style="white-space: nowrap;"|([±φ⁻¹, ±1, ±φ, ±2]) / {{radic|8}} |- |colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[W:Quaternion|quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[W:Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone, Pileio & Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone, Pileio & Levitt|2010|loc=Table 3|p=1442}}}} |} The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}} ===Chords=== [[File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also [[W:Clifford parallel|Clifford parallel]]) and they intersect in a single point. In [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotations]], points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a ''helical'' smooth curve called an ''isocline''{{Efn|An '''isocline''' is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a ''straight line'', a [[W:Geodesic|geodesic]]. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are [[W:Link (knot theory)|linked]] and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} To avoid confusion, we always refer to an ''isocline'' as such, and reserve the term ''[[W:Great circle|great circle]]'' for an ordinary great circle in the plane.|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a [[W:SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has 5-cell edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains an irregular great dodecagon compound of several of these great circle polygons in the same central plane, [[#Compound of five 600-cells|illustrated below]].]] {{see also|600-cell#Golden chords}} The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|[[File:Regular_star_figure_6(5,2).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> six of the 120 disjoint regular 5-cells of edge-length {{radic|2.5}} which are inscribed in the 120-cell appear as six pentagrams, the [[5-cell#Boerdijk–Coxeter helix|Clifford polygon of the 5-cell]]. The 30 vertices comprise a Petrie polygon of the 120-cell,{{Efn|name=two coaxial Petrie 30-gons}} with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In [[600-cell#Decagons and pentadecagrams|600-cell § Decagons and pentadecagrams]], see the illustration of [[W:Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, [[5-cell#Geodesics and rotations|pentagonal circuits]] each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two ''disjoint 5-cells'' are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic [[W:SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>2 disjoint [[W:Pentadecagram|pentadecagram]] isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the ''same'' discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}}]]The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and [[5-cell#Geodesics and rotations|its inscribed regular 5-cells' opposing 1200 edges]].{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an [[#Compound of five 600-cells|irregular great dodecagon]], a compound [[#Chords|great circle polygon of the 120-cell]].|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete [[W:Hopf fibration|Hopf fibration]].{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentadecagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=isocline}} of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} and they are occupied by adjacent [[W:Clifford parallel|Clifford parallel]] great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' rotations.{{Efn|name=distinct rotations}}}} They also give the 120-cell a characteristic great circle polygon: an ''irregular'' great hexagon in which three 120-cell edges alternate with three 5-cell edges.{{Efn|name=irregular great hexagon}} The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the [[5-cell#Geodesics and rotations|5-cell]] and the [[W:8-cell|8-cell tesseract]], they form zig-zag [[W:Petrie polygon|Petrie polygon]]s instead.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical ''isocline''{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its ''Clifford polygon''.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[W:Triacontagon|triacontagon]] {30} zig-zag [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[W:Skew polygon|skew]] regular [[W:Triacontagon|triacontagon]] {30}.{{Efn|name=15 distinct chord lengths}} The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel [[#Compound of five 600-cells|{12} great circle polygons]].]] The 120-cell contains 80 distinct [[W:30-gon|30-gon]] Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons.{{Efn|name=Petrie polygons of the 120-cell}} The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.{{Efn|name=two coaxial Petrie 30-gons}} The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has at least 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the [[W:Petrie polygon|Petrie polygon]] can be picked out as a [[W:Skew polygon|skew polygon]] of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix ring]]. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered [[#Chords|chords]] of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct [[W:Pythagorean distance|Pythagorean distance]]s through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord is the {30/''n''} polygram chord, which connects two vertices that are ''n'' edge lengths apart on a Petrie polygon of the 120-cell. The 15 major chords lie on great circles in central planes that contain regular and irregular polygons of {4}, {10}, or {12} vertices. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). The 15 ''minor chords'' lie on rectangular {4} great circles and do not occur anywhere except inside the 120-cell. In this article, we refer to the 15 minor chords by reference to their arc-angles, e.g. 41.4~° #3⁺ with length {{radic|0.5}} falls between the #3 and #4 chords.|name=additional 120-cell chords}} The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[W:Snub 24-cell|snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=4-polytopes ordered by size and complexity}} The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Great hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[W:Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Great hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[W:H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[W:Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column ''a'' for (30−''i'')₀) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}} |- !colspan=6|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}} ![[5-cell|5-cell]] ![[16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[24-cell|24-cell]] ![[W:Snub 24-cell|Snub]] ![[600-cell]] ![[#Tetrahedrally diminished 120-cell|Dimin]] ! style="border-right: none;"|120-cell ! style="border-left: none;"| |- !colspan=6|Vertices | style="background: seashell;"|5 | style="background: paleturquoise;"|8 | style="background: paleturquoise;"|16 | style="background: paleturquoise;"|24 | style="background: yellow;"|96 | style="background: yellow;"|120 | style="background: seashell;"|480 | style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}} |rowspan=6 style="background: seashell; border: none;"| |- !colspan=6|Edges | style="background: seashell;"|10{{Efn|name=irregular great hexagon}} | style="background: paleturquoise;"|24 | style="background: paleturquoise;"|32 | style="background: paleturquoise;"|96 | style="background: yellow;"|432 | style="background: yellow;"|720 | style="background: seashell;"|1200 | style="background: seashell;"|1200{{Efn|name=irregular great hexagon}} |- !colspan=6|Edge chord | style="background: seashell;{{text color default}};"|#8{{Efn|name=inscribed 5-cells}} | style="background: paleturquoise;"|#7 | style="background: paleturquoise;"|#5 | style="background: paleturquoise;"|#5 | style="background: yellow;"|#3 | style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the [[600-cell]] and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-cell#Boerdijk–Coxeter helix rings|600-cell Petrie polygon is a helical ring]] which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a [[W:Triacontagon#Triacontagram|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a [[600-cell#Decagons|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[600-cell#Boerdijk–Coxeter helix rings|600-cell's Petrie helix]] does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew [[W:Triacontagon#Triacontagram|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[W:Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}} | style="background: seashell;"|#1 | style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}} |- !colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[W:SO(4)#Double rotations|double rotation]] in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its ''rotation angle'' and its ''isocline angle''.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this ''incorrectly'', because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its ''completely'' orthogonal plane, but Clifford parallel to it. It rotates ''with'' its completely orthogonal plane, but not ''in'' it. It is Clifford parallel to its completely orthogonal plane ''and'' to the plane it is moving to, and does not intersect them; the plane that it rotates ''in'' is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, [[16-cell#Rotations|all 6 orthogonal planes]] rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different [[W:Orientation entanglement|orientation]] (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} | style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]] | style="background: paleturquoise;"|[[16-cell#Helical construction|#15]] | style="background: paleturquoise;"|#10 | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]] | style="background: yellow;"|#5 | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]] | style="background: seashell;"|#4 | style="background: seashell;"|#4{{Efn|The characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}} |- !colspan=6|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Efn|name=isoclinic}}|name=Clifford polygon}} | style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]] | style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]] | style="background: paleturquoise;"| | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]] | style="background: yellow;"| | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]] | style="background: seashell;"| | style="background: seashell;"|[[W:Pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}} |- !colspan=3|Chord !Arc !colspan=2|Edge | style="background: seashell;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: yellow;"| | style="background: yellow;"| | style="background: seashell;"| | style="background: seashell;"| |- style="background: seashell;"| |rowspan=2|#1<br>△ |rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]] |rowspan=2|30 |{{Efn|name=120-cell Petrie {30}-gon}} |colspan=2|120-cell edge <big>𝛇</big> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |15.5~° |{{radic|𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <small>{{sfrac|1|φ}}</small> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² = <small>{{sfrac|1|φ²}}</small> ≈ 0.382 {{indent|7}}𝜀 = 𝚫²/2 = <small>{{sfrac|1|2φ⁴}}</small> ≈ 0.073<br> and the 120-cell edge-length is: {{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270<br> For example: {{indent|7}}𝛇 = {{radic|𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}} |0.270~ |- style="background: seashell;"| |rowspan=2|#2<br><big>☐</big> |rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]] |rowspan=2|15 | |colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1₀. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[W:Pentagon#Regular pentagons|pentagon diagonal]] (the #2 chord) is in the [[W:Golden ratio|golden ratio]] φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn|name=dodecahedral cell metrics}}|name=face pentagon chord}}|name=#2 chord}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |25.2~° |{{radic|0.19~}} |0.437~ |- style="background: yellow;"| |rowspan=2|#3<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]] |rowspan=2|10 |𝝅/5 |colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |- style="background: seashell;"| |rowspan=2|#4<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]] |rowspan=2|{{sfrac|15|2}} | |colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |44.5~° |{{radic|0.57~}} |0.757~ |- style="background: paleturquoise;"| |rowspan=2|#5<br>△ |rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]] |rowspan=2|6 |𝝅/3 |colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[W:Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a [[W:0-gon|0-gon]] great circle axis that does not intersect any vertices.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different [[#Compound of five 600-cells|central planes]].|name=great hexagon}} |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2|{{green|<big>'''225'''</big>}}<br><br> |rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |60° |{{radic|1}} |1 |- style="background: yellow;"| |rowspan=2|#6<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]] |rowspan=2|5 |2𝝅/5 |colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |- style="background: paleturquoise;"| |rowspan=2|#7<br><big>☐</big> |rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]] |rowspan=2|{{sfrac|30|7}} |𝝅/2 |colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}} |rowspan=2| |rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24 |rowspan=2|{{green|<big>'''675'''</big>}}<br>48 |rowspan=2|<br>72 |rowspan=2| |rowspan=2|<br>1800 |rowspan=2|<br> |rowspan=2|<br>9000 |rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4} |- style="background: paleturquoise;"| |90° |{{radic|2}} |1.414~ |- style="background: #FFCCCC;"| |rowspan=2|#8<br><big>✩</big> |rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]] |rowspan=2|{{sfrac|15|4}} | |colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[W:Triacontagon|triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}} |rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10 |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: #FFCCCC;"| |104.5~° |{{radic|2.5}} |1.581~ |- style="background: yellow;"| |rowspan=2|#9<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]] |rowspan=2|{{sfrac|10|3}} |3𝝅/5 |colspan=2|[[W:Golden section|golden section]] <math>\phi</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |108° |{{radic|2.𝚽}} |1.618~ |- style="background: paleturquoise;"| |rowspan=2|#10<br>△ |rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]] |rowspan=2|3 |2𝝅/3 |colspan=2|[[24-cell#Great triangles|great triangle]] |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2| |rowspan=2|<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |120° |{{radic|3}} |1.732~ |- style="background: seashell;"| |rowspan=2|#11<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]] |rowspan=2|{{sfrac|30|11}} | |colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: yellow;"| |rowspan=2|#12<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]] |rowspan=2|{{sfrac|5|2}} |4𝝅/5 |colspan=2|great [[W:Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: seashell;"| |rowspan=2|#13<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]] |rowspan=2|{{sfrac|30|13}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/13}-gram]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: seashell;"| |rowspan=2|#14<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]] |rowspan=2|{{sfrac|15|7}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/14}=2{15/7}]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200<br> |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: paleturquoise;"| |rowspan=2|#15<br><small>△☐𝜙</small> |rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]] |rowspan=2|2 |𝝅 |colspan=2|[[W:Diameter|diameter]] |rowspan=2| |rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4 |rowspan=2|<br>8 |rowspan=2|<br>12 |rowspan=2|<br>48 |rowspan=2|<br>60 |rowspan=2|<br>240 |rowspan=2|<br>300{{Efn|name=rays and bases}} |rowspan=2|{{blue|<big>'''1'''</big>}}<br><br> |- style="background: paleturquoise;"| |180° |{{radic|4}} |2 |- !colspan=6|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} | style="background: seashell;"|25 | style="background: paleturquoise;"|64 | style="background: paleturquoise;"|256 | style="background: paleturquoise;"|576 | style="background: yellow;"| | style="background: yellow;"|14400 | style="background: seashell;"| | style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}} !<big>{{blue|'''300'''}}</big> |} [[File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.{{Efn|Drawing the fan of chords with #1 and #11 at a different origin than all the others is an artistic choice, since all the chords are incident at every vertex. We could just as well have drawn all the chords from the same origin vertex, but this arrangement notices the parallel relationship between #8 and #11.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] The annotated chord table is a complete [[W:Bill of materials|bill of materials]] for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the '''#3''' chord row, the 600-cell's 72 great decagons contain 720 '''#3''' chords in all. The '''{{red|red}}''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of <big>{{red|'''25'''}}</big> disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''{{green|green}}''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains <big>{{green|'''225'''}}</big> distinct 24-cells which share components. The '''{{blue|blue}}''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the '''#3''' chord row, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[W:Vertex figure|vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell. === Relationships among interior polytopes === The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, ''except'' for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which ''does'' occur. Various regular [[W:Skew polygon|skew polygon]]s {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional [[W:Jigsaw puzzle|jigsaw puzzle]] in which all those polytopes are the parts.{{Sfn|Schleimer & Segerman|2013}} Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.{{Sfn|Stillwell|2001}} The regular 1-polytope occurs in only [[#Chords|15 distinct lengths]] in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By [[W:Alexandrov's uniqueness theorem|Alexandrov's uniqueness theorem]], convex polyhedra with shapes distinct from each other also have distinct [[W:Metric spaces|metric spaces]] of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four {{background color|paleturquoise|[[24-cell#Hypercubic chords|hypercubic chords]]}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built solely from them. {{see also|W:24-cell#Relationships among interior polytopes|label 1=24-cell § Relationships among interior polytopes}} An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|[[600-cell#Golden chords|golden chords]]}} are square roots of irrational fractions that are functions of {{radic|5}}. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built solely from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. {{see also|W:600-cell#Icosahedra|label 1=600-cell § Icosahedra}} All 15 major chords, and 15 other distinct chordal distances (the minor chords [[120-cell#Geodesic rectangles|enumerated below]]), occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{background color|#FFCCCC|[[5-cell#Boerdijk–Coxeter helix|regular 5-cells and {{radic|5/2}} chords]].}}{{Efn|Dodecahedra emerge as ''visible'' features in the 120-cell, but they also occur in the 600-cell as ''interior'' polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the ''regular'' 5-cell (the [[W:Simplex|simplex]] regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the [[W:Tetrahedron|3-simplex]] and [[W:5-orthoplex|5-orthoplex]]. An [[W:simplex|<math>n</math>-simplex]] is bounded by <math>n+1</math> vertices and <math>n+1</math> (<math>n</math>-1)-simplex facets, and has <math>z+1</math> long diameters (its edges) of length <math>\sqrt{n+1}/\sqrt{n}</math> radii. An [[W:orthoplex|<math>n</math>-orthoplex]] is bounded by <math>2n</math> vertices and <math>2^n</math> (<math>n</math>-1)-simplex facets, and has <math>n</math> long diameters (its orthogonal axes) of length <math>2</math> radii. An [[W:hypercube|<math>n</math>-cube]] is bounded by <math>2^n</math> vertices and <math>2n</math> (<math>n</math>-1)-cube facets, and has <math>2^{n-1}</math> long diameters of length <math>\sqrt{n}</math> radii.{{Efn|The <math>n</math>-simplex's facets are larger than the <math>n</math>-orthoplex's facets. For <math>n=4</math>, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of <math>\sqrt{5}</math> to <math>\sqrt{4}</math> to <math>\sqrt{2}</math>.|name=root 5/root 4/root 2}} The <math>\sqrt{3}</math> long diameters of the 3-cube are shorter than the <math>\sqrt{4}</math> axes of the 3-orthoplex. The [[16-cell#Coordinates|coordinates of the 4-orthoplex]] are the permutations of <math>(0,0,0,\pm 1)</math>, and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of <math>(0,0,0,1)</math>.{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting <math>(0,0,0,1)</math>, and its completely orthogonal 3-facet permuting <math>(0,0,0,-1)</math>, comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-[[W:demihypercube|demicube]], half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to <math>n=4</math>. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} The <math>\sqrt{4}</math> long diameters of the 4-cube are the same length as the <math>\sqrt{4}</math> axes of the 4-orthoplex. The [[W:5-orthoplex#Cartesian coordinates|coordinates of the 5-orthoplex]] are the permutations of <math>(0,0,0,0,\pm 1)</math>, and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of <math>(0,0,0,0,1)</math>.{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting <math>(0,0,0,0,1)</math>, and its completely orthogonal 4-facet permuting <math>(0,0,0,0,-1)</math>, comprise all 10 vertices of the 5-orthoplex.}} The <math>\sqrt{5}</math> long diameters of the 5-cube are longer than the <math>\sqrt{4}</math> axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells.{{Efn|No vertex pair of any of the 120 5-cells (no [[5-cell#Geodesics and rotations|great digon central plane of a 5-cell]]) occurs in any of the 675 16-cells (the 675 [[16-cell#Coordinates|Cartesian basis sets of 6 orthogonal central planes]]).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}} {{see also|W:5-cell#Geodesics and rotations|label 1=5-cell § Geodesics and rotations}} === Compound of five 600-cells === [[File:Great dodecagon of the 120-cell.png|thumb|300px|The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).]] The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} As a consequence of being a compound of five disjoint 600-cells, the 120-cell has 200 irregular great dodecagon {12} central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 {12} central planes originate as the compounds of the hexagonal central planes of the 25 disjoint inscribed 24-cells and the digon central planes of the 120 disjoint inscribed regular 5-cells; they contain all the 24-cell and 5-cell edges, and also the 120-cell edges. Thus the edges and characteristic rotations{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The '''characteristic isoclinic rotation of a 4-polytope''' is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each [[W:Hopf fibration|Hopf fibration]] of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an [[#Compound of five 600-cells|irregular great dodecagon]] which also has #4 chord edges. In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in these same 200 rotation planes.{{Efn|name=edge rotation planes}} Each of the ten 600-cells occupies the entire set of 200 planes. The 120-cell's irregular [[#Other great circle constructs|dodecagon {12} great circle polygon]] has 6 short edges (#1 [[#Chords|chords]] marked {{Color|red|𝜁}}) alternating with 6 longer dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} Inscribed in the irregular great dodecagon are two irregular great hexagons ({{color|red|red}}) in alternate positions.{{Efn|name=irregular great hexagon}} Two ''regular'' great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of 200 dodecagon planes. They occur as 100 completely orthogonal pairs, and they are the ''same'' 200 central planes each containing a [[600-cell#Hexagons|hexagon]] that are found in ''each'' of the 10 inscribed 600-cells. There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares regular hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint regular hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells. ===Concentric hulls=== [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px| Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an Overall Hull that is a [[W:Chamfered dodecahedron|chamfered dodecahedron]] of Norm={{radic|8}}.<br /> Hulls 1 - 8 are the 8 sections of the 120-cell beginning with a cell (Hull 1).<br /> Hulls 1, 2, & 7 are each pairs of [[W:Dodecahedron|dodecahedron]]s.<br /> Hull 3 is a pair of [[W:Icosidodecahedron|icosidodecahedron]]s.<br /> Hulls 4 & 5 are each pairs of [[W:Truncated icosahedron|truncated icosahedron]]s.<br /> Hull 6 is a pair of semi-regular [[W:Rhombicosidodecahedron|rhombicosidodecahedron]]s.<br /> Hull 8 is a single non-uniform [[W:Rhombicosidodecahedron#Names|rhombicosidodecahedron]], the central section.<br /> A more detailed visualization of these 15 simplified sections, with subgroup sections where the inscribed solid has more than one permutation in its orbit, is available [https://commons.wikimedia.org/wiki/File:Cell_First_533_120-Cell_Sections.svg here].]] {{Clear}} These hulls illustrate Coxeter's sections 1₃ - 8₃ of the 120-cell, the sections beginning with a cell (hull #1).{{Sfn|Coxeter|1973|p=299|loc=Table V (iv) Sections of {5,3,3} beginning with a cell (right half of table)}} A ''section'' is a flat 3-dimensional hyperplane slice through the [[W:3-sphere|3-sphere]]: a 2-sphere (ordinary sphere). It is dimensionally analogous to a flat 2-dimensional plane slice through a 2-sphere: a 1-sphere (ordinary circle). The hulls are illustrated as if they were all the same size, but actually they increase in radius as numbered: they are concentric 2-spheres that nest inside each other. Every cell of the 120-cell is the smallest hull in its own set of 8 concentric hulls. There are 120 distinct sets of hulls. The 120-cell actually has 15 sections beginning with a cell, numbered 1 - 15 with number 8 in the center. After increasing in size from 1 to 8, the hulls get smaller again. Sections 1 and 15 are both a hull #1, the smallest hull, a dodecahedral cell of the 120-cell. Section #8 is the central section, the largest hull, with the same radius as the 120-cell. Except for the central section #8, the sections occur in parallel pairs, on either side of the central section. Hull #8 is dimensionally analogous to the equator, while hulls #1 - #7 are dimensionally analogous to lines of latitude. There are 120 of each kind of hull #1 - #7 in the 120-cell, but only 60 of the central hull #8. {{Clear}} The 120-cell also has 30 sections beginning with a vertex, illustrated below. Like the sections beginning with a cell illustrated above, the vertex-first sections are also flat 3-dimensional hyperplane slices through the 3-sphere, and polyhedra that nest inside each other as concentric 2-spheres. Section 0₀ is the vertex itself. Section 1₀ is the 120-cell's tetrahedral vertex figure. Sections 1₀ - 29₀ are described in more detail in [[120-cell#Geodesic rectangles|§Geodesic rectangles]] below. {{Clear}} [[File:Vertex_First_533_120-Cell_Sections.svg|thumb|left|640px| Coxeter's sections 0₀ - 30₀ of the 120-cell, the sections beginning with a vertex, showing the orbit sections and subgroup sections (when the inscribed solid has more than one permutation in its orbit), as well as the convex hull of each orbit on the right.]] {{Clear}} === Geodesic rectangles === The 30 distinct chords{{Efn|name=additional 120-cell chords}} found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Background color|palegreen|△ planes that intersect {12} vertices}} in an [[#Compound of five 600-cells|irregular great dodecagon]], {{Background color|yellow|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and <big>☐</big> planes that intersect {4} vertices in several kinds of {{Background color|gainsboro|rectangle}}, including a {{Background color|seashell|square}}. Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing [[#Concentric hulls|polyhedral sections]] of the 120-cell beginning with a vertex (the 0₀ section), as illustrated above. The correspondence is that each 120-cell vertex is surrounded in curved 3-space <math>S_3</math> by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius in Euclidean 3-space <math>R_3</math>.{{Efn|In the curved 3-dimensional space <math>S_3</math> of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the ''apex'' vertex and its surrounding ''base'' polyhedron form a [[W:Polyhedral pyramid|polyhedral pyramid]]. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} There are 600 distinct sets of 15 hulls. The #1 chord is the radius in <math>S_3</math> of the 1₀ section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the radius in <math>S_3</math> of its congruent opposing 29₀ section. The #7 chord is the radius in <math>S_3</math> of the central vertex-first section of the 120-cell, in which two opposing 15₀ sections are coincident. Each vertex is surrounded by two instances of each polyhedron, at the near and far radial distances of the polyhedron's 180° complementary chords, but because curved space <math>S_3</math> begins to close back up on itself after the #7 90° chord, the near and far concentric polyhedra are the same size. Each chord length is given three ways (on successive lines): for the unit-radius 120-cell as a square root, for the unit-radius 120-cell, and for the unit-edge 120-cell.{{Efn|We give chord lengths as unit-radius square roots in these articles, even when they are integers (e.g. the long diameter is {{radic|4}}). Our usual metric is unit-radius, which reveals relationships among successive 4-polytopes,{{Efn|name=4-polytopes ordered by size and complexity}} but Coxeter{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} and Steinbach{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with "fan of chords" diagrams.|p=23|loc=Figure 3}} use unit-edge, which reveals relationships among successive chords.|name=metrics}} To the left of this last unit-edge metric, its reciprocal^-1 is given. The reciprocal is the long radius of a regular ''n''₀-polygon with unit-radius 120-cell edges (#1 chords) as its edges; but this does not imply that the section ''n''₀ polyhedron contains any ''n''₀ polygons.{{Efn|The 120-cell contains no regular {30} central polygons, although its Petrie polygon is a skew regular {30}. Therefore the edge of the regular triacontagon {30} is not a chord of the 120-cell represented in this table. Nevertheless these metrics of the {30} are relevant:<br> :Unit-radius {30}: ::Edge <small><math>E = 2 \sin{\pi/30} \approx \sqrt{0.0437} \approx 0.209</math></small> :Unit-edge {30}: ::Radius <small><math>R_{ue} = 1/E \approx 4.783</math></small> :{30} with 120-cell edges: ::Edge <small><math>\zeta \approx 0.270~</math></small> ::<small><math>E \approx 0.774 \times \zeta</math></small> ::Radius <small><math>R_\zeta \approx 1.292</math></small> |name=triacontagon metrics}} {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |#0<br><br>0₀ | |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |<math>\pi</math> |{{radic|4}} |{{radic|4}} | rowspan="3" |#15<br><br>30₀ |- style="background: palegreen;" | |0° |0 |0 |180° |2 |2 |- style="background: palegreen;" | | |0 |<small><math>0\times\zeta</math></small> |0.135~^-1 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#1<br><br>1₀ |𝞯 |{{radic|0.𝜀}}{{Efn|name=fractional square roots}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} | |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |#14<br><br>29₀ |- style="background: palegreen;" | |15.5~°{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}} |0.270~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |164.5~° |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |1^-1 |1 |<small><math>1\times\zeta</math></small> |0.136~^-1 |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#2<br><br>2₀ |{{Efn|name=#2 chord}} |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 | |{{radic|3.81~}} | | rowspan="3" |#13<br><br>28₀ |- style="background: gainsboro;" | |25.2~° |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |154.8~° |1.952~ | |- style="background: gainsboro;" | |0.618~^-1 |1.618~ |<small><math>\phi\times\zeta</math></small> |0.138~^-1 |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |#3<br><br>3₀ |<math>\pi / 5</math> |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |<math>4\pi / 5</math> |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |#12<br><br>27₀ |- style="background: yellow;" | |36° |0.618~ |<small><math>1 / \phi</math></small> |144°{{Efn|name=dihedral}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.437~^-1 |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |0.142~^-1 |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#3⁺<br><br>4₀ | |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |#12⁻<br><br>26₀ |- style="background: gainsboro;" | |41.4~° |0.707~ |<small><math>\sqrt{2}/2</math></small> |138.6~° |1.871~ | |- style="background: gainsboro;" | |0.382~^-1 |2.618~ |<small><math>\phi^2\times\zeta</math></small> |0.144~^-1 |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#4<br><br>5₀ | |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown). Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' great hexagons (the [[600-cell#Hexagons|24-cell's characteristic rotation]]), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in ''irregular'' great hexagons).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every ''regular'' great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.|name=#4 isocline chord bridge}}|name=dodecagon rotation}}<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" |{{Efn|name=#4 isocline chord bridge}} | |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |#11<br><br>25₀ |- style="background: palegreen;" | |44.5~° |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |135.5~° |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.357~^-1 |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |0.146~^-1 |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#4⁺<br><br>6₀ | |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |#11⁻<br><br>24₀ |- style="background: gainsboro;" | |49.1~° |0.831~ | |130.9~° |1.819~ | |- style="background: gainsboro;" | |0.325~^-1 |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |0.148~^-1 |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁻<br><br>7₀ | |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |#10⁺<br><br>23₀ |- style="background: gainsboro;" | |56° |0.939~ | |124° |1.766~ | |- style="background: gainsboro;" | |0.288~^-1 |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |0.153~^-1 |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |#5<br><br>8₀ |<math>\pi / 3</math> |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}}<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |<small><math>2\pi / 3</math></small> |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |#10<br><br>22₀ |- style="background: palegreen;" | |60° |1 | |120° |1.732~ | |- style="background: palegreen;" | |0.270~^-1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |0.156~^-1 |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁺<br><br>9₀ | |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |#10⁻<br><br>21₀ |- style="background: gainsboro;" | |66.1~° |1.091~ | |113.9~° |1.676~ | |- style="background: gainsboro;" | |0.247~^-1 |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |0.161~^-1 |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁻<br><br>10₀ | |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |#9⁺<br><br>20₀ |- style="background: gainsboro;" | |69.8~° |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |110.2~° |1.640~ | |- style="background: gainsboro;" | |0.236~^-1 |4.236~ |<small><math>\phi^3\times\zeta</math></small> |0.165~^-1 |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |#6<br><br>11₀ |<math>2\pi/5</math> |{{radic|1.𝚫}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}}<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |<math>3\pi / 5</math> |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |#9<br><br>19₀ |- style="background: yellow;" | |72° |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |108° |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |0.230~^-1 |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |0.167~^-1 |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |#6⁺⁻<br><br>12₀ | |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic|2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12₀ sections and 18₀ sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}}<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 | |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |#8<br><br>18₀ |- style="background: palegreen;" | |75.5~° |1.224~ | |104.5~° |1.581~ | |- style="background: palegreen;" | |0.221~^-1 |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |0.171~^-1 |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺<br><br>13₀ | |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.31~}} | | rowspan="3" |#8⁻<br><br>17₀ |- style="background: gainsboro;" | |81.1~° |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |98.9~° |1.520~ | |- style="background: gainsboro;" | |0.208~⁻¹ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |0.178~^-1 |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺⁺<br><br>14₀ | |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |#7⁺<br><br>16₀ |- style="background: gainsboro;" | |84.5~° |1.345~ | |95.5~° |1.480~ | |- style="background: gainsboro;" | |0.201~⁻¹ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |0.182~^-1 |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |#7<br><br>15₀ |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}}<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |#7<br><br>15₀ |- style="background: seashell;" | |90° |1.414~ | |90° |1.414~ | |- style="background: seashell;" | |0.191~⁻¹ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |0.191~^-1 |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|name=isoclinic rotation}} of a distinct class,{{Efn|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons" / 4.}} In each class of rotation,{{Efn|[[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]] are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are ''invariant'', which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually [[W:Clifford parallel|Clifford parallel]]. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called ''fibrations'', which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular ''fibers'', the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} ===Polyhedral graph=== Considering the [[W:Adjacency matrix|adjacency matrix]] of the vertices representing the polyhedral graph of the unit-radius 120-cell, the [[W:Graph diameter|graph diameter]] is 15, connecting each vertex to its coordinate-negation at a [[W:Euclidean distance|Euclidean distance]] of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are [[W:Vertex coloring|3-colorable]]. The graph is [[W:Eulerian path|Eulerian]] having degree 4 in every vertex. Its edge set can be decomposed into two [[W:Hamiltonian path|Hamiltonian cycles]].<ref>{{cite book| author = Carlo H. Séquin | title = Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes | date = July 2005 | pages = 463–472 | publisher = Mathartfun.com | isbn = 9780966520163 | url = https://archive.bridgesmathart.org/2005/bridges2005-463.html#gsc.tab=0 | access-date=March 13, 2023}}</ref> === Constructions === The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the [[600-cell]],{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the [[24-cell|24-cell]],{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 24-cell}} the 24-cell can be deconstructed into three distinct instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 '''rays''' [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a '''[[W:Orthonormal basis|basis]]'''. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 300₉675₄ to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct [[16-cell#Coordinates|16-cell]] containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse [[W:Golden ratio|golden ratio]]), but the 120-cell's edge length is ~0.270 times its radius. The 120-cell is also the convex hull of the regular compound of 120 disjoint regular 5-cells. This can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point 600-cell, expand each vertex into a regular 5-cell. For each of the 120 vertices, add 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point 600-cells: a 120-cell. ==== Dual 600-cells ==== [[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]] Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ({{sfrac|φ²|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ²{{radic|2}} ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The [[#Unit radius coordinates|unit radius 120-cell]] (with edge-length {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ²}} ≈ 1.080. [[File:Dodecahedron_vertices.svg|thumb|150px|right|One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.{{Efn|In the [[W:120-cell#Dual 600-cells|dodecahedral cell]] of the unit-radius 120-cell, the edge is the '''15.5° #1 [[#Chords|chord]]''' of the 120-cell of length <small><math>\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270</math></small>. Eight {{Color|orange}} vertices lie at the Cartesian coordinates <small><math>(\pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8})</math></small> relative to origin at the cell center. They form a cube (dashed lines) whose edges are the '''25.2° #2 chord''' of length <small><math>\tfrac{1}{\phi\sqrt{2}} \approx 0.437</math></small> (the pentagon diagonal). The face diagonals of the cube (not drawn) are the '''36° #3 chord''' of length <small><math>\tfrac{1}{\phi} \approx 0.618</math></small> (the edges of two 600-cell tetrahedron cells inscribed in the cube). The next largest '''41.1° chord''' has length <small><math>\tfrac{1}{\sqrt{2}} \approx 0.707</math></small>. The diameter of the dodecahedron is the '''44.5° #4 chord''' of length <small><math>\tfrac{\sqrt{3}}{\phi\sqrt{2}} \approx 0.757</math></small> (the cube diagonal). If the #4 diameter is extended outside the dodecahedron in a straight line in the curved space of the 3-sphere, it is colinear with a #1 edge belonging to three neighboring dodecahedron cells, and the combined '''60° #5 chord''' has length <small><math>\sqrt{1}</math></small> (an edge of an inscribed 24-cell). If this 60° combined #4 plus #1 geodesic is further extended in a straight line by another #4 chord (the diameter of a further cell), the combined '''104.5° #8 chord''' has length <small><math>\tfrac{\sqrt{5}}{\sqrt{2}} \approx 1.581</math></small> (an edge of an inscribed regular 5-cell).|name=dodecahedral cell metrics}}]] Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ²|{{radic|8}}}} ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given [[#√8 radius coordinates|above]] of long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ²}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ², which is smaller than {{Radic|8}} in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ²{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps. ==== Cell rotations of inscribed duals ==== Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} This shows that the 120-cell contains, among its many interior features, 120 [[W:Compound of ten tetrahedra|compounds of ten tetrahedra]], each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because [[#Cell rotations of inscribed duals|each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells]]. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 600-cells}} All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-cell.}} Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. ==== Augmentation ==== Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing [[W:Hyperpyramid|4-pyramid]]s of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a [[W:Regular dodecahedron#Cartesian coordinates|dodecahedron]].{{Efn|name=truncated apex}} Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a [[600-cell#Icosahedra|cluster of five tetrahedra]], with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.{{Efn|As we saw in the [[600-cell#Cell clusters|600-cell]], these 12 tetrahedra belong (in pairs) to the 6 [[600-cell#Icosahedra|icosahedral clusters]] of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. ==== Weyl orbits ==== Another construction method uses [[W:Quaternion|quaternion]]s and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe <math>T</math> and <math>T'</math> [[24-cell|24-cell]]s as quaternion orbit weights of D4 under the Weyl group W(D4):<br/> O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}<br/> O(1000) : V1<br/> O(0010) : V2<br/> O(0001) : V3 <math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math> With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the 120-cell of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the [[600-cell]] <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> * the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math> * the [[W:Dual snub 24-cell|dual snub 24-cell]] = <math>T \oplus T' \oplus S'</math>. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}} <math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} ! [[W:K-face|''k''-face]]||f_k||f₀ || f₁||f₂||f₃||[[W:vertex figure|''k''-fig]] !Notes |- align=right |A₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|3|node}} ||( ) !f₀ || 600 || 4 || 6 || 4 ||[[W:Regular tetrahedron|{3,3}]] || H₄/A₃ = 14400/24 = 600 |- align=right |A₁A₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|3|node}} ||{ } !f₁ || 2 || 1200 || 3 || 3 || [[W:Equilateral triangle|{3}]] || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |H₂A₁ ||{{Coxeter–Dynkin diagram|node_1|5|node|2|node_x|2|node}} ||[[W:Pentagon|{5}]] !f₂ || 5 || 5 || 720 || 2 || { } || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |H₃ ||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|2|node_x}} ||[[W:Regular dodecahedron|{5,3}]] !f₃ || 20 || 30 || 12 ||120|| ( ) || H₄/H₃ = 14400/120 = 120 |} == Visualization == The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[24-cell|24-cell]]). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}} === Layered stereographic projection === The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer & Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12). Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an [[W:Icosidodecahedron|icosidodecahedron]], with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="4" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 12 cells | First layer of meridional cells / "[[W:Arctic Circle|Arctic Circle]]" | style="text-align: center" | 36° |- | style="text-align: center" | 3 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 60° |- | style="text-align: center" | 4 | style="text-align: center" | 12 cells | Second layer of meridional cells / "[[W:Tropic of Cancer|Tropic of Cancer]]" | style="text-align: center" | 72° |- | style="text-align: center" | 5 | style="text-align: center" | 30 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" | Equator |- | style="text-align: center" | 6 | style="text-align: center" | 12 cells | Third layer of meridional cells / "[[W:Tropic of Capricorn|Tropic of Capricorn]]" | style="text-align: center" | 108° | rowspan="4" | Southern Hemisphere |- | style="text-align: center" | 7 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 120° |- | style="text-align: center" | 8 | style="text-align: center" | 12 cells | Fourth layer of meridional cells / "[[W:Antarctic Circle|Antarctic Circle]]" | style="text-align: center" | 144° |- | style="text-align: center" | 9 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 120 cells ! colspan="3" | |} The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell. === Intertwining rings === [[Image:120-cell rings.jpg|right|thumb|300px|Two intertwining rings of the 120-cell.]] [[File:120-cell_two_orthogonal_rings.png|thumb|300px|Two orthogonal rings in a cell-centered projection]] The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized [[W:Hopf fibration|Hopf fibration]].{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer & Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical [[W:Torsion of a curve|torsion]]. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two ([[W:Clifford torus|Clifford]]) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B²|pp=23-29}} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint ([[W:Clifford parallel|Clifford parallel]]) great circles. === Other great circle constructs === There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter [[#Chords|chords]], forming an [[#Compound of five 600-cells|irregular dodecagon in a central plane]]. Both these great circle paths have dual [[600-cell#Union of two tori|great circle paths in the 600-cell]]. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a [[600-cell#Decagons|decagon]].{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six [[W:Triangular bipyramids|triangular bipyramids]]) in the 600-cell. This latter path corresponds to a [[600-cell#Icosahedra|ring of six icosahedra]] meeting face to face in the [[W:Snub 24-cell|snub 24-cell]] (or [[W:Icosahedral pyramid|icosahedral pyramids]] in the 600-cell), forming a [[600-cell#Hexagons|hexagon]]. Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges [[#Chords|illustrated above]].{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each [[#Compound of five 600-cells|irregular great dodecagon]], in alternate positions. === 2D Orthogonal projections === [[W:Orthographic projection|Orthogonal projection]]s of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by [[W:B. L. Chilton|B. L. Chilton]].{{Sfn|Chilton|1964}} The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:Van Oss polygon|van Oss polygon]]. {| class="wikitable" |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:120-cell graph H4.svg|240px]]<br>[30]<br>(Red=1) |[[File:120-cell t0 p20.svg|240px]]<br>[20]<br>(Red=1) |[[File:120-cell t0 F4.svg|240px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:120-cell t0 H3.svg|240px]]<br>[10]<br>(Red=5, orange=10) |[[File:120-cell t0 A2.svg|240px]]<br>[6]<br>(Red=1, orange=3, yellow=6, lime=9, green=12) |[[File:120-cell t0 A3.svg|240px]]<br>[4]<br>(Red=1, orange=2, yellow=4, lime=6, green=8) |} === 3D Perspective projections === These projections use [[W:Perspective projection|perspective projection]], from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. A comparison of perspective projections of the 3D dodecahedron to 2D (above left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. [[W:Schlegel diagram|Schlegel diagram]]s use [[W:Perspective (graphical)|perspective]] to show depth in the dimension which has been flattened, choosing a view point ''above'' a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. [[W:Stereographic projection|Stereographic projection]]s use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a [[W:3-sphere|3-sphere]]. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating [[120-cell#Animations|animations]] below, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must). {| class="wikitable" style="width:540px;" |+Comparison with regular dodecahedron |- !width=80|Projection ![[W:Dodecahedron|Dodecahedron]] !120-cell |- ![[W:Schlegel diagram|Schlegel diagram]] |align=center|[[Image:Dodecahedron schlegel.svg|220px]]<br>12 pentagon faces in the plane |align=center|[[File:Schlegel wireframe 120-cell.png|220px]]<br>120 dodecahedral cells in 3-space |- ![[W:Stereographic projection|Stereographic projection]] |align=center|[[Image:Dodecahedron stereographic projection.png|220px]] |align=center|[[Image:Stereographic polytope 120cell faces.png|220px]]<br>With transparent faces |} {|class="wikitable" |- !colspan=2|Enhanced perspective projections |- |align=center|[[Image:120-cell perspective-cell-first-02.png|240px]] |Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: * Nearest dodecahedron to the 4D viewpoint rendered in yellow * The 12 dodecahedra immediately adjoining it rendered in cyan; * The remaining dodecahedra rendered in green; * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image. |- |align=center|[[Image:120-cell perspective-vertex-first-02.png|240px]] |Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: * Four cells surrounding nearest vertex shown in 4 colors * Nearest vertex shown in white (center of image where 4 cells meet) * Remaining cells shown in transparent green * Cells facing away from 4D viewpoint culled for clarity |} === Animations === {|class="wikitable" !colspan=2|Projections to 3D of a 4D 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]] |- |align=center|[[File:120-cell.gif|256px]] |align=center|[[File:120-cell-inner.gif|256px]] |- |From outside the [[W:3-sphere|3-sphere]] in 4-space. |Inside the [[600-cell#Boundary envelopes|3D surface]] of the 3-sphere. |} In all the above projections of the 120-cell, only the edges of the 120-cell appear. All the other [[#Chords|chords]] are not shown. 600 chords converge at ''each'' of the 600 vertices. The complex [[#Relationships among interior polytopes|interior parts]] of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.{{Efn|[[File:Omnitruncated_120-cell_Coxeter_sections-subsections_projected_from_4D.svg|thumb|A full display of each section's orbits along with sub-section orbits in the 14400-point omnitruncated 120-cell.]]The 120-cell has <small><math>600^2 = 360,000</math></small> distinct chords. With all of its chords ''and their intersections'' it is the 14400 vertex [[W:Omnitruncation|omnitruncated]] 120-cell, which is identical to the omnitruncated 600-cell given the symmetry of their Coxeter-Dynkin diagrams.}} The following animation is an exception which does show some interior chords, although it does not reveal the inscribed 4-polytopes. {| class=wikitable width=540 !colspan=1|Coxeter section views |- |align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]<br>Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls. |} == Related polyhedra and honeycombs== === H₄ polytopes === The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} {{H4_family}} === {p,3,3} polytopes === The 120-cell is similar to three [[W:Regular 4-polytope|regular 4-polytopes]]: the [[5-cell|5-cell]] {3,3,3} and [[W:Tesseract|tesseract]] {4,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[W:Tetrahedral|tetrahedral]] [[W:Vertex figure|vertex figure]] {3,3}: {{Tetrahedral vertex figure tessellations small}} === {5,3,p} polytopes === The 120-cell is a part of a sequence of 4-polytopes and honeycombs with [[W:Dodecahedral|dodecahedral]] cells: {{Dodecahedral_tessellations_small}} === Tetrahedrally diminished 120-cell === Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.{{Efn|The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the [[600-cell#Diminished 600-cells|diminishment of the 120-point 600-cell]] by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point [[W:Snub 24-cell|snub 24-cell]]. Similarly, the 8-cell tesseract can be seen as a 16-point [[24-cell#Diminishings|diminished 24-cell]] from which one 8-point 16-cell has been removed.}} [[File:Tetrahedrally_diminished_regular_dodecahedron.png|thumb|In the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]], 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.]] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]] because the 4 vertices removed formed a [[#Dual 600-cells|tetrahedron inscribed in the dodecahedron]]. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.{{Efn|name=face pentagon chord}} The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron,{{Efn|Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular [[5-cell|5-cell]] with a regular tetrahedron base.|name=truncated apex}} each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges. The '''480-point diminished 120-cell''' may be called the '''tetrahedrally diminished 120-cell''' because its cells are tetrahedrally diminished, or the '''600-cell diminished 120-cell''' because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the '''regular 5-cells diminished 120-cell''' because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.{{Efn|name=inscribed 5-cells}} === Davis 120-cell manifold === The '''Davis 120-cell manifold''', introduced by {{harvtxt|Davis|1985}}, is a compact 4-dimensional [[W:Hyperbolic manifold|hyperbolic manifold]] obtained by identifying opposite faces of the 120-cell, whose universal cover gives the [[W:List of regular polytopes#Tessellations of hyperbolic 4-space|regular honeycomb]] [[W:order-5 120-cell honeycomb|{5,3,3,5}]] of 4-dimensional hyperbolic space. ==See also== *[[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] *[[W:57-cell|57-cell]] – an abstract regular 4-polytope constructed from 57 [[W:Hemi-dodecahedron|hemi-dodecahedra]]. *[[600-cell]] - the dual [[W:4-polytope|4-polytope]] to the 120-cell ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} * {{Citation | last1=Davis | first1=Michael W. | title=A hyperbolic 4-manifold | doi=10.2307/2044771 | year=1985 | journal=[[W:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=93 | issue=2 | pages=325–328| jstor=2044771 }} *[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} * {{Cite journal|last1=Schleimer|first1=Saul|last2=Segerman|first2=Henry|date=2013|title=Puzzling the 120-cell|journal=Notices Amer. Math. Soc.|volume=62|issue=11|pages=1309–1316|doi=10.1090/noti1297 |arxiv=1310.3549 |s2cid=117636740|ref={{SfnRef|Schleimer & Segerman|2013}}}} {{Refend}} ==External links== * [https://www.youtube.com/watch?v=MFXRRW9goTs/ YouTube animation of the construction of the 120-cell] Gian Marco Todesco. * [http://www.theory.org/geotopo/120-cell/ Construction of the Hyper-Dodecahedron] * [http://www.gravitation3d.com/120cell/ 120-cell explorer] &ndash; A free interactive program (requires Microsoft .Net framework) that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL. [[Category:Geometry]] [[Category:Polyscheme]] 6t0v5n52qgc3o3czkmbb3j8wa92ai08 2806691 2806690 2026-04-26T18:41:45Z Dc.samizdat 2856930 /* Concentric hulls */ 2806691 wikitext text/x-wiki {{Short description|Four-dimensional analog of the dodecahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=120-cell | Image_File=Schlegel wireframe 120-cell.png | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]<br>(vertices and edges) | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]] | Last=[[W:Snub 24-cell|31]] | Index=32 | Next=[[W:Rectified 120-cell|33]] | Schläfli={5,3,3}| CD={{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}}| Cell_List=120 [[W:Dodecahedron|{5,3}]] [[Image:Dodecahedron.png|20px]]| Face_List=720 [[W:Pentagon|{5}]] [[File:Regular pentagon.svg|20px]]| Edge_Count=1200| Vertex_Count= 600| Petrie_Polygon=[[W:Triacontagon|30-gon]]| Coxeter_Group=H₄, [3,3,5]| Vertex_Figure=[[File:120-cell verf.svg|80px]]<br>[[W:Tetrahedron|tetrahedron]]| Dual=[[600-cell]]| Property_List=[[W:Convex set|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]] }} {{maths}} [[File:120-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[W:Geometry|geometry]], the '''120-cell''' is the [[W:Convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {5,3,3}. It is also called a '''C₁₂₀''', '''dodecaplex''' (short for "dodecahedral complex"), '''hyperdodecahedron''', '''polydodecahedron''', '''hecatonicosachoron''', '''dodecacontachoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hecatonicosahedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> The boundary of the 120-cell is composed of 120 dodecahedral [[W:Cell (mathematics)|cells]] with 4 meeting at each vertex. Together they form 720 [[W:Pentagon|pentagonal]] faces, 1200 edges, and 600 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:Regular dodecahedron|regular dodecahedron]], since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the ''dodecaplex'' has 120 dodecahedral facets, with 3 around each edge.{{Efn|In the 120-cell, 3 dodecahedra and 3 pentagons meet at every edge. 4 dodecahedra, 6 pentagons, and 4 edges meet at every vertex. The dihedral angle (between dodecahedral hyperplanes) is 144°.{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}}|name=dihedral}} Its dual polytope is the [[600-cell]]. == Geometry == The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).{{Efn|name=elements}} As the sixth and largest regular convex 4-polytope,{{Efn|name=4-polytopes ordered by size and complexity}} it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the [[5-cell|5-cell]],{{Efn|name=inscribed 5-cells}} which is not found in any of the others.{{Sfn|Dechant|2021|p=18|loc=''Remark 5.7''|ps=, explains why not.{{Efn|name=rotated 4-simplexes are completely disjoint}}}} The 120-cell is a four-dimensional [[W:Swiss Army knife|Swiss Army knife]]: it contains one of everything. It is daunting but instructive to study the 120-cell, because it contains examples of ''every'' relationship among ''all'' the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} That is why [[W:John Stillwell|Stillwell]] titled his paper on the 4-polytopes and the history of mathematics<ref>''Mathematics and Its History'', John Stillwell, 1989, 3rd edition 2010, {{isbn|0-387-95336-1}}</ref> of more than 3 dimensions ''The Story of the 120-cell''.{{Sfn|Stillwell|2001}} {{Regular convex 4-polytopes|wiki=W:|radius=1}} ===Cartesian coordinates=== Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen. ==== √8 radius coordinates ==== The 120-cell with long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 has edge length 4−2φ = 3−{{radic|5}} ≈ 0.764. In this frame of reference, its 600 vertex coordinates are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} of the following:{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} {| class=wikitable |- !24 | ({0, 0, ±2, ±2}) | [[24-cell#Great squares|24-cell]] | rowspan=7 | 600-point 120-cell |- !64 | ({±φ, ±φ, ±φ, ±φ⁻²}) | |- !64 | ({±1, ±1, ±1, ±{{radic|5}}<nowiki />}) | |- !64 | ({±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²}) | |- !96 | ([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !96 | ([0, ±φ⁻², ±1, ±φ²]) | [[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !192 | ([±φ⁻¹, ±1, ±φ, ±2]) | |} where φ (also called 𝝉){{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}} is the [[W:Golden ratio|golden ratio]], {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618. ==== Unit radius coordinates ==== The unit-radius 120-cell has edge length {{Sfrac|1|φ²{{Radic|2}}}} ≈ 0.270. In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinates{{Sfn|Mamone, Pileio & Levitt|2010|p=1442|loc=Table 3}} are the {[[W:Permutations|permutations]]} and {{bracket|[[W:Even permutation|even permutation]]s}} in the left column below: {| class="wikitable" style=width:720px |- !rowspan=3|120 !8 |style="white-space: nowrap;"|({±1, 0, 0, 0}) |[[16-cell#Coordinates|16-cell]] | rowspan="2" |[[24-cell#Great hexagons|24-cell]] | rowspan="3" |[[600-cell#Coordinates|600-cell]] | rowspan="10" style="white-space: nowrap;"|120-cell |- !16 |style="white-space: nowrap;"|({±1, ±1, ±1, ±1}) / 2 |[[W:Tesseract#Radial equilateral symmetry|Tesseract]] |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±1, ±φ]) / 2 |colspan=2|[[W:Snub 24-cell#Coordinates|Snub 24-cell]] |- !rowspan=7|480 !colspan=2|[[#Tetrahedrally diminished 120-cell|Diminished 120-cell]] !5-point [[5-cell#Coordinates|5-cell]] ![[24-cell#Great squares|24-cell]] ![[600-cell#Coordinates|600-cell]] |- !32 |style="white-space: nowrap;"|([±φ, ±φ, ±φ, ±φ⁻²]) / {{radic|8}} |rowspan=6 style="white-space: nowrap;"|(1, 0, 0, 0)<br> (−1,{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},−{{radic|5}},{{spaces|2}}{{radic|5}}) / 4<br> (−1,−{{radic|5}},{{spaces|2}}{{radic|5}},−{{radic|5}}) / 4<br> (−1,{{spaces|2}}{{radic|5}},−{{radic|5}},−{{radic|5}}) / 4 |rowspan=6 style="white-space: nowrap;"|({±{{radic|1/2}}, ±{{radic|1/2}}, 0, 0}) |rowspan=6 style="white-space: nowrap;"|({±1, 0, 0, 0})<br> ({±1, ±1, ±1, ±1}) / 2<br> ([0, ±φ⁻¹, ±1, ±φ]) / 2 |- !32 |style="white-space: nowrap;"|([±1, ±1, ±1, ±{{radic|5}}]) / {{radic|8}} |- !32 |style="white-space: nowrap;"|([±φ⁻¹, ±φ⁻¹, ±φ⁻¹, ±φ²]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻¹, ±φ, ±{{radic|5}}]) / {{radic|8}} |- !96 |style="white-space: nowrap;"|([0, ±φ⁻², ±1, ±φ²]) / {{radic|8}} |- !192 |style="white-space: nowrap;"|([±φ⁻¹, ±1, ±φ, ±2]) / {{radic|8}} |- |colspan=7|The unit-radius coordinates of uniform convex 4-polytopes are related by [[W:Quaternion|quaternion]] multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible [[W:Quaternion#Multiplication of basis elements|quaternion products]]{{Sfn|Mamone, Pileio & Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).{{Efn|To obtain all 600 coordinates by quaternion cross-multiplication of these three 4-polytopes' coordinates with less redundancy, it is sufficient to include just one vertex of the 24-cell: ({{radic|1/2}}, {{radic|1/2}}, 0, 0).{{Sfn|Mamone, Pileio & Levitt|2010|loc=Table 3|p=1442}}}} |} The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.{{Efn|The 120-cell can be constructed as a compound of '''{{red|5}}''' disjoint 600-cells,{{Efn|name=2 ways to get 5 disjoint 600-cells}} or '''{{red|25}}''' disjoint 24-cells, or '''{{red|75}}''' disjoint 16-cells, or '''{{red|120}}''' disjoint 5-cells. Except in the case of the 120 5-cells,{{Efn|Multiple instances of each of the regular convex 4-polytopes can be inscribed in any of their larger successor 4-polytopes, except for the smallest, the regular 5-cell, which occurs inscribed only in the largest, the 120-cell.{{Efn|name=simplex-orthoplex-cube relation}} To understand the way in which the 4-polytopes nest within each other, it is necessary to carefully distinguish ''disjoint'' multiple instances from merely ''distinct'' multiple instances of inscribed 4-polytopes. For example, the 600-point 120-cell is the convex hull of a compound of 75 8-point 16-cells that are completely disjoint: they share no vertices, and 75 * 8 {{=}} 600. But it is also possible to pick out 675 distinct 16-cells within the 120-cell, most pairs of which share some vertices, because two concentric equal-radius 16-cells may be rotated with respect to each other such that they share 2 vertices (an axis), or even 4 vertices (a great square plane), while their remaining vertices are not coincident.{{Efn|name=rays and bases}} In 4-space, any two congruent regular 4-polytopes may be concentric but rotated with respect to each other such that they share only a common subset of their vertices. Only in the case of the 4-simplex (the 5-point regular 5-cell) that common subset of vertices must always be empty, unless it is all 5 vertices. It is impossible to rotate two concentric 4-simplexes with respect to each other such that some, but not all, of their vertices are coincident: they may only be completely coincident, or completely disjoint. Only the 4-simplex has this property; the 16-cell, and by extension any larger regular 4-polytope, may lie rotated with respect to itself such that the pair shares some, but not all, of their vertices. Intuitively we may see how this follows from the fact that only the 4-simplex does not possess any opposing vertices (any 2-vertex central axes) which might be invariant after a rotation. The 120-cell contains 120 completely disjoint regular 5-cells, which are its only distinct inscribed regular 5-cells, but every other nesting of regular 4-polytopes features some number of disjoint inscribed 4-polytopes and a larger number of distinct inscribed 4-polytopes.|name=rotated 4-simplexes are completely disjoint}} these are not counts of ''all'' the distinct regular 4-polytopes which can be found inscribed in the 120-cell, only the counts of ''completely disjoint'' inscribed 4-polytopes which when compounded form the convex hull of the 120-cell. The 120-cell contains '''{{green|10}}''' distinct 600-cells, '''{{green|225}}''' distinct 24-cells, and '''{{green|675}}''' distinct 16-cells.{{Efn|name=rays and bases}}|name=inscribed counts}} ===Chords=== [[File:Great polygons of the 120-cell.png|thumb|300px|Great circle polygons of the 120-cell, which lie in the invariant central planes of its isoclinic{{Efn|Two angles are required to specify the separation between two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)".}} If the two angles are identical, the two planes are called isoclinic (also [[W:Clifford parallel|Clifford parallel]]) and they intersect in a single point. In [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotations]], points rotate within invariant central planes of rotation by some angle, and the entire invariant central plane of rotation also tilts sideways (in an orthogonal invariant central plane of rotation) by some angle. Therefore each vertex traverses a ''helical'' smooth curve called an ''isocline''{{Efn|An '''isocline''' is a closed, curved, helical great circle through all four dimensions. Unlike an ordinary great circle it does not lie in a single central plane, but like any great circle, when viewed within the curved 3-dimensional space of the 4-polytope's boundary surface it is a ''straight line'', a [[W:Geodesic|geodesic]]. Both ordinary great circles and isocline great circles are helical in the sense that parallel bundles of great circles are [[W:Link (knot theory)|linked]] and spiral around each other, but neither are actually twisted (they have no inherent torsion). Their curvature is not their own, but a property of the 3-sphere's natural curvature, within which curved space they are finite (closed) straight line segments.{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles. An ordinary great circle is an isocline of circumference <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} To avoid confusion, we always refer to an ''isocline'' as such, and reserve the term ''[[W:Great circle|great circle]]'' for an ordinary great circle in the plane.|name=isocline}} between two points in different central planes, while traversing an ordinary great circle in each of two orthogonal central planes (as the planes tilt relative to their original planes). If the two orthogonal angles are identical, the distance traveled along each great circle is the same, and the double rotation is called isoclinic (also a [[W:SO(4)#Isoclinic rotations|Clifford displacement]]). A rotation which takes isoclinic central planes to each other is an isoclinic rotation.{{Efn|name=isoclinic rotation}}|name=isoclinic}} rotations. The 120-cell edges of length {{Color|red|𝜁}} ≈ 0.270 occur only in the {{Color|red|red}} irregular great hexagon, which also has 5-cell edges of length {{Color|red|{{radic|2.5}}}}. The 120-cell's 1200 edges do not form great circle polygons by themselves, but by alternating with {{radic|2.5}} edges of inscribed regular 5-cells{{Efn|name=inscribed 5-cells}} they form 400 irregular great hexagons.{{Efn|name=irregular great hexagon}} The 120-cell also contains an irregular great dodecagon compound of several of these great circle polygons in the same central plane, [[#Compound of five 600-cells|illustrated below]].]] {{see also|600-cell#Golden chords}} The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.{{Efn|[[File:Regular_star_figure_6(5,2).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/12}=6{5/2}]],<br> six of the 120 disjoint regular 5-cells of edge-length {{radic|2.5}} which are inscribed in the 120-cell appear as six pentagrams, the [[5-cell#Boerdijk–Coxeter helix|Clifford polygon of the 5-cell]]. The 30 vertices comprise a Petrie polygon of the 120-cell,{{Efn|name=two coaxial Petrie 30-gons}} with 30 zig-zag edges (not shown), and 3 inscribed great decagons (edges not shown) which lie Clifford parallel to the projection plane.{{Efn|Inscribed in the 3 Clifford parallel great decagons of each helical Petrie polygon of the 120-cell{{Efn|name=inscribed 5-cells}} are 6 great pentagons{{Efn|In [[600-cell#Decagons and pentadecagrams|600-cell § Decagons and pentadecagrams]], see the illustration of [[W:Triacontagon#Triacontagram|triacontagram {30/6}=6{5}]].}} in which the 6 pentagrams (regular 5-cells) appear to be inscribed, but the pentagrams are skew (not parallel to the projection plane); each 5-cell actually has vertices in 5 different decagon-pentagon central planes in 5 completely disjoint 600-cells.|name=great pentagon}}]]Inscribed in the unit-radius 120-cell are 120 disjoint regular 5-cells,{{Sfn|Coxeter|1973|loc=Table VI (iv): 𝐈𝐈 = {5,3,3}|p=304}} of edge-length {{radic|2.5}}. No regular 4-polytopes except the 5-cell and the 120-cell contain {{radic|2.5}} chords (the #8 chord).{{Efn|name=rotated 4-simplexes are completely disjoint}} The 120-cell contains 10 distinct inscribed 600-cells which can be taken as 5 disjoint 600-cells two different ways. Each {{radic|2.5}} chord connects two vertices in disjoint 600-cells, and hence in disjoint 24-cells, 8-cells, and 16-cells.{{Efn|name=simplex-orthoplex-cube relation}} Both the 5-cell edges and the 120-cell edges connect vertices in disjoint 600-cells. Corresponding polytopes of the same kind in disjoint 600-cells are Clifford parallel and {{radic|2.5}} apart. Each 5-cell contains one vertex from each of 5 disjoint 600-cells.{{Efn|The 120 regular 5-cells are completely disjoint. Each 5-cell contains two distinct Petrie pentagons of its #8 edges, [[5-cell#Geodesics and rotations|pentagonal circuits]] each binding 5 disjoint 600-cells together in a distinct isoclinic rotation characteristic of the 5-cell. But the vertices of two ''disjoint 5-cells'' are not linked by 5-cell edges, so each distinct circuit of #8 chords is confined to a single 5-cell, and there are no other circuits of 5-cell edges (#8 chords) in the 120-cell.|name=distinct circuits of the 5-cell}}.|name=inscribed 5-cells}} These two additional chords give the 120-cell its characteristic [[W:SO(4)#Isoclinic rotations|isoclinic rotation]],{{Efn|[[File:Regular_star_figure_2(15,4).svg|thumb|200px|In [[W:Triacontagon#Triacontagram|triacontagram {30/8}=2{15/4}]],<br>2 disjoint [[W:Pentadecagram|pentadecagram]] isoclines are visible: a black and a white isocline (shown here as orange and faint yellow) of the 120-cell's characteristic isoclinic rotation.{{Efn|Each black or white pentadecagram isocline acts as both a right isocline in a distinct right isoclinic rotation and as a left isocline in a distinct left isoclinic rotation, but isoclines do not have inherent chirality.{{Efn|name=isocline}} No isocline is both a right and left isocline of the ''same'' discrete left-right rotation (the same fibration).}} The pentadecagram edges are #4 chords{{Efn|name=#4 isocline chord}} joining vertices which are 8 vertices apart on the 30-vertex circumference of this projection, the zig-zag Petrie polygon.{{Efn|name=pentadecagram isoclines}}]]The characteristic isoclinic rotation{{Efn|name=characteristic rotation}} of the 120-cell takes place in the invariant planes of its 1200 edges{{Efn|name=non-planar geodesic circle}} and [[5-cell#Geodesics and rotations|its inscribed regular 5-cells' opposing 1200 edges]].{{Efn|The invariant central plane of the 120-cell's characteristic isoclinic rotation{{Efn|name=120-cell characteristic rotation}} contains an irregular great hexagon {6} with alternating edges of two different lengths: 3 120-cell edges of length 𝜁 {{=}} {{radic|𝜀}} (#1 chords), and 3 inscribed regular 5-cell edges of length {{radic|2.5}} (#8 chords). These are, respectively, the shortest and longest edges of any regular 4-polytope. {{Efn|Each {{radic|2.5}} chord is spanned by 8 zig-zag edges of a Petrie 30-gon,{{Efn|name=120-cell Petrie {30}-gon}} none of which lie in the great circle of the irregular great hexagon. Alternately the {{radic|2.5}} chord is spanned by 9 zig-zag edges, one of which (over its midpoint) does lie in the same great circle.{{Efn|name=irregular great hexagon}}|name=spanned by 8 or 9 edges}} Each irregular great hexagon lies completely orthogonal to another irregular great hexagon.{{Efn|name=perpendicular and parallel}} The 120-cell contains 400 distinct irregular great hexagons (200 completely orthogonal pairs), which can be partitioned into 100 disjoint irregular great hexagons (a discrete fibration of the 120-cell) in four different ways. Each fibration has its distinct left (and right) isoclinic rotation in 50 pairs of completely orthogonal invariant central planes. Two irregular great hexagons occupy the same central plane, in alternate positions, just as two great pentagons occupy a great decagon plane. The two irregular great hexagons form an [[#Compound of five 600-cells|irregular great dodecagon]], a compound [[#Chords|great circle polygon of the 120-cell]].|name=irregular great hexagon}} There are four distinct characteristic right (and left) isoclinic rotations, each left-right pair corresponding to a discrete [[W:Hopf fibration|Hopf fibration]].{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of [[5-cell#Geodesics and rotations|pentadecagram isoclinic rotations of an individual 5-cell]]; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation.}} In each rotation all 600 vertices circulate on helical isoclines of 15 vertices, following a geodesic circle{{Efn|name=isocline}} with 15 chords that form a {15/4} pentadecagram.{{Efn|The characteristic isocline{{Efn|name=isocline}} of the 120-cell is a skew pentadecagram of 15 #4 chords. Successive #4 chords of each pentadecagram lie in different △ central planes which are inclined isoclinically to each other at 12°, which is 1/30 of a great circle (but not the arc of a 120-cell edge, the #1 chord).{{Efn|name=12° rotation angle}} This means that the two planes are separated by two equal 12° angles,{{Efn|name=isoclinic}} and they are occupied by adjacent [[W:Clifford parallel|Clifford parallel]] great polygons (irregular great hexagons) whose corresponding vertices are joined by oblique #4 chords. Successive vertices of each pentadecagram are vertices in completely disjoint 5-cells. Each pentadecagram is a #4 chord-path{{Efn|name=non-planar geodesic circle}} visiting 15 vertices belonging to three different 5-cells. The two pentadecagrams shown in the {30/8}{{=}}2{15/4} projection{{Efn|name=120-cell characteristic rotation}} visit the six 5-cells that appear as six disjoint pentagrams in the {30/12}{{=}}6{5/2} projection.{{Efn|name=inscribed 5-cells}}|name=pentadecagram isoclines}}|name=120-cell characteristic rotation}} in addition to all the rotations of the other regular 4-polytopes which it inherits.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄|pp=1438-1439|ps=; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct ''isoclinic'' rotations.{{Efn|name=distinct rotations}}}} They also give the 120-cell a characteristic great circle polygon: an ''irregular'' great hexagon in which three 120-cell edges alternate with three 5-cell edges.{{Efn|name=irregular great hexagon}} The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the [[5-cell#Geodesics and rotations|5-cell]] and the [[W:8-cell|8-cell tesseract]], they form zig-zag [[W:Petrie polygon|Petrie polygon]]s instead.{{Efn|The 5-cell, 8-cell and 120-cell all have tetrahedral vertex figures. In a 4-polytope with a tetrahedral vertex figure, a path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. In the 120-cell the 30-edge circumferential path along edges follows a zig-zag skew Petrie polygon, which is not a great circle. However, there exists a 15-chord circumferential path that is a true geodesic great circle through those 15 vertices: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is a helical ''isocline''{{Efn|name=isocline}} that bends in a circle in two completely orthogonal central planes at once, circling through four dimensions rather than confined to a two dimensional plane.{{Efn|name=pentadecagram isoclines}} The skew chord set of an isocline is called its ''Clifford polygon''.{{Efn|name=Clifford polygon}}|name=non-planar geodesic circle}} The [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|120-cell's Petrie polygon]] is a [[W:Triacontagon|triacontagon]] {30} zig-zag [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].{{Efn|[[File:Regular polygon 30.svg|thumb|200px|The Petrie polygon of the 120-cell is a [[W:Skew polygon|skew]] regular [[W:Triacontagon|triacontagon]] {30}.{{Efn|name=15 distinct chord lengths}} The 30 #1 chord edges do not all lie on the same {30} great circle polygon, but they lie in groups of 6 (equally spaced around the circumference) in 5 Clifford parallel [[#Compound of five 600-cells|{12} great circle polygons]].]] The 120-cell contains 80 distinct [[W:30-gon|30-gon]] Petrie polygons of its 1200 edges, and can be partitioned into 20 disjoint 30-gon Petrie polygons.{{Efn|name=Petrie polygons of the 120-cell}} The Petrie 30-gon twists around its 0-gon great circle axis 9 times in the course of one circular orbit, and can be seen as a compound [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] of 600-cell edges (#3 chords) linking pairs of vertices that are 9 vertices apart on the Petrie polygon.{{Efn|name=two coaxial Petrie 30-gons}} The {30/9}-gram (with its #3 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #1 chord edges).|name=120-cell Petrie {30}-gon}} Since the 120-cell has a circumference of 30 edges, it has at least 15 distinct chord lengths, ranging from its edge length to its diameter.{{Efn|The 30-edge circumference of the 120-cell follows a skew Petrie polygon, not a great circle polygon. The Petrie polygon of any 4-polytope is a zig-zag helix spiraling through the curved 3-space of the 4-polytope's surface.{{Efn|The Petrie polygon of a 3-polytope (polyhedron) with triangular faces (e.g. an icosahedron) can be seen as a linear strip of edge-bonded faces bent into a ring. Within that circular strip of edge-bonded triangles (10 in the case of the icosahedron) the [[W:Petrie polygon|Petrie polygon]] can be picked out as a [[W:Skew polygon|skew polygon]] of edges zig-zagging (not circling) through the 2-space of the polyhedron's surface: alternately bending left and right, and slaloming around a great circle axis that passes through the triangles but does not intersect any vertices. The Petrie polygon of a 4-polytope (polychoron) with tetrahedral cells (e.g. a 600-cell) can be seen as a linear helix of face-bonded cells bent into a ring: a [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix ring]]. Within that circular helix of face-bonded tetrahedra (30 in the case of the 600-cell) the skew Petrie polygon can be picked out as a helix of edges zig-zagging (not circling) through the 3-space of the polychoron's surface: alternately bending left and right, and spiraling around a great circle axis that passes through the tetrahedra but does not intersect any vertices.}} The 15 numbered [[#Chords|chords]] of the 120-cell occur as the distance between two vertices in that 30-vertex helical ring.{{Efn|name=additional 120-cell chords}} Those 15 distinct [[W:Pythagorean distance|Pythagorean distance]]s through 4-space range from the 120-cell edge-length which links any two nearest vertices in the ring (the #1 chord), to the 120-cell axis-length (diameter) which links any two antipodal (most distant) vertices in the ring (the #15 chord).|name=15 distinct chord lengths}} Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the additional chords occur only in the interior of 120-cell, not as edges of any of the six regular convex 4-polytopes or their characteristic great circle rings. The 15 ''[[#Chords|major chords]]'' are so numbered because the #''n'' chord is the {30/''n''} polygram chord, which connects two vertices that are ''n'' edge lengths apart on a Petrie polygon of the 120-cell. The 15 major chords lie on great circles in central planes that contain regular and irregular polygons of {4}, {10}, or {12} vertices. There are [[#Geodesic rectangles|30 distinct 4-space chordal distances]] between vertices of the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). The 15 ''minor chords'' lie on rectangular {4} great circles and do not occur anywhere except inside the 120-cell. In this article, we refer to the 15 minor chords by reference to their arc-angles, e.g. 41.4~° #3⁺ with length {{radic|0.5}} falls between the #3 and #4 chords.|name=additional 120-cell chords}} The first thing to notice about this table is that it has eight columns, not six; in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point [[W:Snub 24-cell|snub 24-cell]] and the 480-point [[#Tetrahedrally diminished 120-cell|diminished 120-cell]].{{Efn|name=4-polytopes ordered by size and complexity}} The second thing to notice is that each numbered row (each chord) is marked with a triangle <small>△</small>, square ☐, phi symbol 𝜙 or pentagram ✩. The 15 chords form polygons of four kinds: great squares ☐ [[16-cell#Coordinates|characteristic of the 16-cell]], great hexagons and great triangles △ [[24-cell#Great hexagons|characteristic of the 24-cell]], great decagons and great pentagons 𝜙 [[600-cell#Hopf spherical coordinates|characteristic of the 600-cell]], and skew pentagrams ✩ [[5-cell#Geodesics and rotations|characteristic of the 5-cell]] which circle through a set of central planes and form face polygons but not great polygons.{{Efn|The {{radic|2}} edges and 4𝝅 characteristic rotations{{Efn|name=isocline circumference}} of the [[16-cell#Coordinates|16-cell]] lie in the great square ☐ central planes; rotations of this type are an expression of the [[W:Hyperoctahedral group|symmetry group <math>B_4</math>]]. The {{radic|1}} edges, {{radic|3}} chords and 4𝝅 characteristic rotations of the [[24-cell#Great hexagons|24-cell]] lie in the great triangle (great hexagon) △ central planes; rotations of this type are an expression of the [[W:F4 (mathematics)|<math>F_4</math>]] symmetry group. The edges and 5𝝅 characteristic rotations of the [[600-cell#Hopf spherical coordinates|600-cell]] lie in the great pentagon (great decagon) 𝜙 central planes; these chords are functions of {{radic|5}}, and rotations of this type are an expression of the [[W:H4 polytope|symmetry group <math>H_4</math>]]. The polygons and characteristic rotations of the regular [[5-cell#Geodesics and rotations|5-cell]] do not lie in a single central plane; they describe a skew pentagram ✩ or larger skew polygram and only form face polygons, not central polygons; rotations of this type are expressions of the [[W:Tetrahedral symmetry|<math>A_4</math>]] symmetry group.|name=edge rotation planes}} {| class=wikitable style="white-space:nowrap;text-align:center" !colspan=15|Chords of the 120-cell and its inscribed 4-polytopes{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex|ps=; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀, polyhedra whose successively increasing "radii" on the 3-sphere (in column 2''la'') are the following chords in our notation:{{Efn|name=additional 120-cell chords}} #1, #2, #3, 41.4~°, #4, 49.1~°, 56.0~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, ..., #15. The remaining distinct chords occur as the longer "radii" of the second set of 16 opposing polyhedral sections (in column ''a'' for (30−''i'')₀) which lists #15, #14, #13, #12, 138.6~°, #11, 130.1~°, 124~°, #10, 113.9~°, 110.2~°, #9, #8, 98.9~°, 95.5~°, #7, 84.5~°, ..., or at least they occur among the 180° complements of all those Coxeter-listed chords. The complete ordered set of 30 distinct chords is 0°, #1, #2, #3, 41.4~°, #4, 49.1~°, 56~°, #5, 66.1~°, 69.8~°, #6, 75.5~°, 81.1~°, 84.5~°, #7, 95.5~°, #8, #9, 110.2°, 113.9°, #10, 124°, 130.1°, #11, 138.6°, #12, #13, #14, #15. The chords also occur among the edge-lengths of the polyhedral sections (in column 2''lb'', which lists only: #2, .., #3, .., 69.8~°, .., .., #3, .., .., #5, #8, .., .., .., #7, ... because the multiple edge-lengths of irregular polyhedral sections are not given).}} |- !colspan=6|Inscribed{{Efn|"At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in completely orthogonal subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself.... In fact, the [various] radii ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope}}|name=Coxeter on orthogonal dual pairs}} ![[5-cell|5-cell]] ![[16-cell|16-cell]] ![[W:8-cell|8-cell]] ![[24-cell|24-cell]] ![[W:Snub 24-cell|Snub]] ![[600-cell]] ![[#Tetrahedrally diminished 120-cell|Dimin]] ! style="border-right: none;"|120-cell ! style="border-left: none;"| |- !colspan=6|Vertices | style="background: seashell;"|5 | style="background: paleturquoise;"|8 | style="background: paleturquoise;"|16 | style="background: paleturquoise;"|24 | style="background: yellow;"|96 | style="background: yellow;"|120 | style="background: seashell;"|480 | style="background: seashell; border-right: none;"|600{{Efn|name=rays and bases}} |rowspan=6 style="background: seashell; border: none;"| |- !colspan=6|Edges | style="background: seashell;"|10{{Efn|name=irregular great hexagon}} | style="background: paleturquoise;"|24 | style="background: paleturquoise;"|32 | style="background: paleturquoise;"|96 | style="background: yellow;"|432 | style="background: yellow;"|720 | style="background: seashell;"|1200 | style="background: seashell;"|1200{{Efn|name=irregular great hexagon}} |- !colspan=6|Edge chord | style="background: seashell;{{text color default}};"|#8{{Efn|name=inscribed 5-cells}} | style="background: paleturquoise;"|#7 | style="background: paleturquoise;"|#5 | style="background: paleturquoise;"|#5 | style="background: yellow;"|#3 | style="background: yellow;"|#3{{Efn|[[File:Regular_star_figure_3(10,3).svg|180px|thumb|In [[W:Triacontagon#Triacontagram|triacontagram {30/9}{{=}}3{10/3}]] we see the 120-cell Petrie polygon (on the circumference of the 30-gon, with 120-cell edges not shown) as a compound of three Clifford parallel 600-cell great decagons (seen as three disjoint {10/3} decagrams) that spiral around each other. The 600-cell edges (#3 chords) connect vertices which are 3 600-cell edges apart (on a great circle), and 9 120-cell edges apart (on a Petrie polygon). The three disjoint {10/3} great decagons of 600-cell edges delineate a single [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix 30-tetrahedron ring]] of an inscribed 600-cell.]] The 120-cell and 600-cell both have 30-gon Petrie polygons.{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the [[600-cell]] and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[600-cell#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]] (the Petrie polygons of the 600-cell):{{Efn|[[File:Regular_star_polygon_30-11.svg|180px|thumb|The Petrie polygon of the inscribed 600-cells can be seen in this projection to the plane of a triacontagram {30/11}, a 30-gram of #11 chords. The 600-cell Petrie is a helical ring which winds around its own axis 11 times. This projection along the axis of the ring cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with #11 chords connecting every 11th vertex on the circle. The 600-cell edges (#3 chords) which are the Petrie polygon edges are not shown in this illustration, but they could be drawn around the circumference, connecting every 3rd vertex.]]The [[600-cell#Boerdijk–Coxeter helix rings|600-cell Petrie polygon is a helical ring]] which twists around its 0-gon great circle axis 11 times in the course of one circular orbit. Projected to the plane completely orthogonal to the 0-gon plane, the 600-cell Petrie polygon can be seen to be a [[W:Triacontagon#Triacontagram|triacontagram {30/11}]] of 30 #11 chords linking pairs of vertices that are 11 vertices apart on the circumference of the projection.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The {30/11}-gram (with its #11 chord edges) is an alternate sequence of the same 30 vertices as the Petrie 30-gon (with its #3 chord edges).|name={30/11}-gram}} connecting their 30 tetrahedral cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a [[600-cell#Decagons|discrete fibration of the 600-cell]]).{{Efn|name=two coaxial Petrie 30-gons}}|name=Petrie polygons of the 120-cell}} They are two distinct skew 30-gon helices, composed of 30 120-cell edges (#1 chords) and 30 600-cell edges (#3 chords) respectively, but they occur in completely orthogonal pairs that spiral around the same 0-gon great circle axis. The 120-cell's Petrie helix winds closer to the axis than the [[600-cell#Boerdijk–Coxeter helix rings|600-cell's Petrie helix]] does, because its 30 edges are shorter than the 600-cell's 30 edges (and they zig-zag at less acute angles). A dual pair{{Efn|name=Petrie polygons of the 120-cell}} of these Petrie helices of different radii sharing an axis do not have any vertices in common; they are completely disjoint.{{Efn|name=Coxeter on orthogonal dual pairs}} The 120-cell Petrie helix (versus the 600-cell Petrie helix) twists around the 0-gon axis 9 times (versus 11 times) in the course of one circular orbit, forming a skew [[W:Triacontagon#Triacontagram|{30/9}{{=}}3{10/3} polygram]] (versus a skew [[W:Triacontagon#Triacontagram|{30/11} polygram]]).{{Efn|name={30/11}-gram}}|name=two coaxial Petrie 30-gons}} | style="background: seashell;"|#1 | style="background: seashell;"|#1{{Efn|name=120-cell Petrie {30}-gon}} |- !colspan=6|[[600-cell#Rotations on polygram isoclines|Isocline chord]]{{Efn|An isoclinic{{Efn|name=isoclinic}} rotation is an equi-rotation-angled [[W:SO(4)#Double rotations|double rotation]] in two completely orthogonal invariant central planes of rotation at the same time. Every discrete isoclinic rotation has two characteristic arc-angles (chord lengths), its ''rotation angle'' and its ''isocline angle''.{{Efn|name=characteristic rotation}} In each incremental rotation step from vertex to neighboring vertex, each invariant rotation plane rotates by the rotation angle, and also tilts sideways (like a coin flipping) by an equal rotation angle.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} Thus each vertex rotates on a great circle by one rotation angle increment, while simultaneously the whole great circle rotates with the completely orthogonal great circle by an equal rotation angle increment.{{Efn|It is easiest to visualize this ''incorrectly'', because the completely orthogonal great circles are Clifford parallel and do not intersect (except at the central point). Neither do the invariant plane and the plane it moves to. An invariant plane tilts sideways in an orthogonal central plane which is not its ''completely'' orthogonal plane, but Clifford parallel to it. It rotates ''with'' its completely orthogonal plane, but not ''in'' it. It is Clifford parallel to its completely orthogonal plane ''and'' to the plane it is moving to, and does not intersect them; the plane that it rotates ''in'' is orthogonal to all these planes and intersects them all.{{Efn|The plane in which an entire invariant plane rotates (tilts sideways) is (incompletely) orthogonal to both completely orthogonal invariant planes, and also Clifford parallel to both of them.{{Efn|Although perpendicular and linked (like adjacent links in a taught chain), completely orthogonal great polygons are also parallel, and lie exactly opposite each other in the 4-polytope, in planes that do not intersect except at one point, the common center of the two linked circles.|name=perpendicular and parallel}}}} In the 120-cell's characteristic rotation,{{Efn|name=120-cell characteristic rotation}} each invariant rotation plane is Clifford parallel to its completely orthogonal plane, but not adjacent to it; it reaches some other (nearest) parallel plane first. But if the isoclinic rotation taking it through successive Clifford parallel planes is continued through 90°, the vertices will have moved 180° and the tilting rotation plane will reach its (original) completely orthogonal plane.{{Efn|The 90 degree isoclinic rotation of two completely orthogonal planes takes them to each other. In such a rotation of a rigid 4-polytope, [[16-cell#Rotations|all 6 orthogonal planes]] rotate by 90 degrees, and also tilt sideways by 90 degrees to their completely orthogonal (Clifford parallel) plane.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} The corresponding vertices of the two completely orthogonal great polygons are {{radic|4}} (180°) apart; the great polygons (Clifford parallel polytopes) are {{radic|4}} (180°) apart; but the two completely orthogonal ''planes'' are 90° apart, in the ''two'' orthogonal angles that separate them.{{Efn|name=isoclinic}} If the isoclinic rotation is continued through another 90°, each vertex completes a 360° rotation and each great polygon returns to its original plane, but in a different [[W:Orientation entanglement|orientation]] (axes swapped): it has been turned "upside down" on the surface of the 4-polytope (which is now "inside out"). Continuing through a second 360° isoclinic rotation (through four 90° by 90° isoclinic steps, a 720° rotation) returns everything to its original place and orientation.|name=exchange of completely orthogonal planes}}|name=rotating with the completely orthogonal rotation plane}} The product of these two simultaneous and equal great circle rotation increments is an overall displacement of each vertex by the isocline angle increment (the isocline chord length). Thus the rotation angle measures the vertex displacement in the reference frame of a moving great circle, and also the sideways displacement of the moving great circle (the distance between the great circle polygon and the adjacent Clifford parallel great circle polygon the rotation takes it to) in the stationary reference frame. The isocline chord length is the total vertex displacement in the stationary reference frame, which is an oblique chord between the two great circle polygons (the distance between their corresponding vertices in the rotation).|name=isoclinic rotation}} | style="background: seashell;"|[[5-cell#Geodesics and rotations|#8]] | style="background: paleturquoise;"|[[16-cell#Helical construction|#15]] | style="background: paleturquoise;"|#10 | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|#10]] | style="background: yellow;"|#5 | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|#5]] | style="background: seashell;"|#4 | style="background: seashell;"|#4{{Efn|The characteristic isoclinic rotation of the 120-cell, in the invariant planes in which its edges (#1 chords) lie, takes those edges to similar edges in Clifford parallel central planes. Since an isoclinic rotation{{Efn|name=isoclinic rotation}} is a double rotation (in two completely orthogonal invariant central planes at once), in each incremental rotation step from vertex to neighboring vertex the vertices travel between central planes on helical great circle isoclines, not on ordinary great circles,{{Efn|name=isocline}} over an isocline chord which in this particular rotation is a #4 chord of 44.5~° arc-length.{{Efn|The isocline chord of the 120-cell's characteristic rotation{{Efn|name=120-cell characteristic rotation}} is the #4 chord of 44.5~° arc-angle (the larger edge of the irregular great dodecagon), because in that isoclinic rotation by two equal 12° rotation angles{{Efn|name=12° rotation angle}} each vertex moves to another vertex 4 edge-lengths away on a Petrie polygon, and the circular geodesic path it rotates on (its isocline){{Efn|name=isocline}} does not intersect any nearer vertices.|name=120-cell rotation angle}}|name=#4 isocline chord}} |- !colspan=6|Clifford polygon{{Efn|The chord-path of an isocline{{Efn|name=isocline}} may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygram shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Efn|name=isoclinic}}|name=Clifford polygon}} | style="background: seashell;"|[[5-cell#Boerdijk–Coxeter helix|{5/2}]] | style="background: paleturquoise;"|[[16-cell#Helical construction|{8/3}]] | style="background: paleturquoise;"| | style="background: paleturquoise;"|[[24-cell#Helical hexagrams and their isoclines|{6/2}]] | style="background: yellow;"| | style="background: yellow;"|[[600-cell#Decagons and pentadecagrams|{15/2}]] | style="background: seashell;"| | style="background: seashell;"|[[W:Pentadecagram|{15/4}]]{{Efn|name=120-cell characteristic rotation}} |- !colspan=3|Chord !Arc !colspan=2|Edge | style="background: seashell;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: paleturquoise;"| | style="background: yellow;"| | style="background: yellow;"| | style="background: seashell;"| | style="background: seashell;"| |- style="background: seashell;"| |rowspan=2|#1<br>△ |rowspan=2|[[File:Regular_polygon_30.svg|50px|{30}]] |rowspan=2|30 |{{Efn|name=120-cell Petrie {30}-gon}} |colspan=2|120-cell edge <big>𝛇</big> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{red|<big>'''1'''</big>}}<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |15.5~° |{{radic|𝜀}}{{Efn|1=The fractional square root chord lengths are given as decimal fractions where: {{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <small>{{sfrac|1|φ}}</small> {{indent|7}}𝚫 = 1 - 𝚽 = 𝚽² = <small>{{sfrac|1|φ²}}</small> ≈ 0.382 {{indent|7}}𝜀 = 𝚫²/2 = <small>{{sfrac|1|2φ⁴}}</small> ≈ 0.073<br> and the 120-cell edge-length is: {{indent|7}}𝛇 = {{radic|𝜀}} = {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270<br> For example: {{indent|7}}𝛇 = {{radic|𝜀}} = {{radic|0.073~}} ≈ 0.270|name=fractional square roots|group=}} |0.270~ |- style="background: seashell;"| |rowspan=2|#2<br><big>☐</big> |rowspan=2|[[File:Regular_star_figure_2(15,1).svg|50px|{30/2}=2{15}]] |rowspan=2|15 | |colspan=2|face diagonal{{Efn|The #2 chord joins vertices which are 2 edge lengths apart: the vertices of the 120-cell's tetrahedral vertex figure, the second section of the 120-cell beginning with a vertex, denoted 1₀. The #2 chords are the edges of this tetrahedron, and the #1 chords are its long radii. The #2 chords are also diagonal chords of the 120-cell's pentagon faces.{{Efn|The face [[W:Pentagon#Regular pentagons|pentagon diagonal]] (the #2 chord) is in the [[W:Golden ratio|golden ratio]] φ ≈ 1.618 to the face pentagon edge (the 120-cell edge, the #1 chord).{{Efn|name=dodecahedral cell metrics}}|name=face pentagon chord}}|name=#2 chord}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |25.2~° |{{radic|0.19~}} |0.437~ |- style="background: yellow;"| |rowspan=2|#3<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,1).svg|50px|{30/3}=3{10}]] |rowspan=2|10 |𝝅/5 |colspan=2|[[600-cell#Decagons|great decagon]] <math>\phi^{-1}</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|{{green|<big>'''10'''</big>}}{{Efn|name=inscribed counts}}<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |36° |{{radic|0.𝚫}} |0.618~ |- style="background: seashell;"| |rowspan=2|#4<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,2).svg|50px|{30/4}=2{15/2}]] |rowspan=2|{{sfrac|15|2}} | |colspan=2|cell diameter{{Efn||name=dodecahedral cell metrics}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |44.5~° |{{radic|0.57~}} |0.757~ |- style="background: paleturquoise;"| |rowspan=2|#5<br>△ |rowspan=2|[[File:Regular_star_figure_5(6,1).svg|50px|{30/5}=5{6}]] |rowspan=2|6 |𝝅/3 |colspan=2|[[600-cell#Hexagons|great hexagon]]{{Efn|[[File:Regular_star_figure_5(6,1).svg|thumb|180px|[[W:Triacontagon#Triacontagram|Triacontagram {30/5}=5{6}]], the 120-cell's skew Petrie 30-gon as a compound of 5 great hexagons.]] Each great hexagon edge is the axis of a zig-zag of 5 120-cell edges. The 120-cell's Petrie polygon is a helical zig-zag of 30 120-cell edges, spiraling around a [[W:0-gon|0-gon]] great circle axis that does not intersect any vertices.{{Efn|name=two coaxial Petrie 30-gons}} There are 5 great hexagons inscribed in each Petrie polygon, in five different [[#Compound of five 600-cells|central planes]].|name=great hexagon}} |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{green|<big>'''225'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2|{{green|<big>'''225'''</big>}}<br><br> |rowspan=2|{{red|<big>'''5'''</big>}}{{Efn|name=inscribed counts}}<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |60° |{{radic|1}} |1 |- style="background: yellow;"| |rowspan=2|#6<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,1).svg|50px|{30/6}=6{5}]] |rowspan=2|5 |2𝝅/5 |colspan=2|[[600-cell#Decagons and pentadecagrams|great pentagon]]{{Efn|name=great pentagon}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |72° |{{radic|1.𝚫}} |1.175~ |- style="background: paleturquoise;"| |rowspan=2|#7<br><big>☐</big> |rowspan=2|[[File:Regular_star_polygon_30-7.svg|50px|{30/7}]] |rowspan=2|{{sfrac|30|7}} |𝝅/2 |colspan=2|[[600-cell#Squares|great square]]{{Efn|name=rays and bases}} |rowspan=2| |rowspan=2|{{green|<big>'''675'''</big>}}{{Efn|name=rays and bases}}<br>24 |rowspan=2|{{green|<big>'''675'''</big>}}<br>48 |rowspan=2|<br>72 |rowspan=2| |rowspan=2|<br>1800 |rowspan=2|<br> |rowspan=2|<br>9000 |rowspan=2|{{blue|<big>'''54'''</big>}}<br>9{3,4} |- style="background: paleturquoise;"| |90° |{{radic|2}} |1.414~ |- style="background: #FFCCCC;"| |rowspan=2|#8<br><big>✩</big> |rowspan=2|[[File:Regular_star_figure_2(15,4).svg|50px|{30/8}=2{15/4}]] |rowspan=2|{{sfrac|15|4}} | |colspan=2|[[5-cell#Boerdijk–Coxeter helix|5-cell]]{{Efn|The [[5-cell#Boerdijk–Coxeter helix|Petrie polygon of the 5-cell]] is the pentagram {5/2}. The Petrie polygon of the 120-cell is the [[W:Triacontagon|triacontagon]] {30}, and one of its many projections to the plane is the triacontagram {30/12}{{=}}6{5/2}.{{Efn|name=120-cell Petrie {30}-gon}} Each 120-cell Petrie 6{5/2}-gram lies completely orthogonal to six 5-cell Petrie {5/2}-grams, which belong to six of the 120 disjoint regular 5-cells inscribed in the 120-cell.{{Efn|name=inscribed 5-cells}}|name=orthogonal Petrie polygons}} |rowspan=2|{{red|<big>'''120'''</big>}}{{Efn|name=inscribed 5-cells}}<br>10 |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2|<br>1200{{Efn|name=120-cell characteristic rotation}} |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: #FFCCCC;"| |104.5~° |{{radic|2.5}} |1.581~ |- style="background: yellow;"| |rowspan=2|#9<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_3(10,3).svg|50px|{30/9}=3{10/3}]] |rowspan=2|{{sfrac|10|3}} |3𝝅/5 |colspan=2|[[W:Golden section|golden section]] <math>\phi</math> |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |108° |{{radic|2.𝚽}} |1.618~ |- style="background: paleturquoise;"| |rowspan=2|#10<br>△ |rowspan=2|[[File:Regular_star_figure_10(3,1).svg|50px|{30/10}=10{3}]] |rowspan=2|3 |2𝝅/3 |colspan=2|[[24-cell#Great triangles|great triangle]] |rowspan=2| |rowspan=2| |rowspan=2|<br>32 |rowspan=2|{{red|<big>'''25'''</big>}}{{Efn|name=inscribed counts}}<br>96 |rowspan=2| |rowspan=2|<br>1200 |rowspan=2| |rowspan=2|<br>2400 |rowspan=2|{{blue|<big>'''32'''</big>}}<br>4{4,3} |- style="background: paleturquoise;"| |120° |{{radic|3}} |1.732~ |- style="background: seashell;"| |rowspan=2|#11<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-11.svg|50px|{30/11}]] |rowspan=2|{{sfrac|30|11}} | |colspan=2|[[600-cell#Boerdijk–Coxeter helix rings|{30/11}-gram]]{{Efn|name={30/11}-gram}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200 |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |135.5~° |{{radic|3.43~}} |1.851~ |- style="background: yellow;"| |rowspan=2|#12<br><big>𝜙</big> |rowspan=2|[[File:Regular_star_figure_6(5,2).svg|50px|{30/12}=6{5/2}]] |rowspan=2|{{sfrac|5|2}} |4𝝅/5 |colspan=2|great [[W:Pentagon#Regular pentagons|pent diag]]{{Efn|name=orthogonal Petrie polygons}} |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>720 |rowspan=2| |rowspan=2|<br>7200 |rowspan=2|{{blue|<big>'''24'''</big>}}<br>2{3,5} |- style="background: yellow;"| |144°{{Efn|name=dihedral}} |{{radic|3.𝚽}} |1.902~ |- style="background: seashell;"| |rowspan=2|#13<br><big>✩</big> |rowspan=2|[[File:Regular_star_polygon_30-13.svg|50px|{30/13}]] |rowspan=2|{{sfrac|30|13}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/13}-gram]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>3600<br> |rowspan=2|{{blue|<big>'''12'''</big>}}<br>2{3,4} |- style="background: seashell;"| |154.8~° |{{radic|3.81~}} |1.952~ |- style="background: seashell;"| |rowspan=2|#14<br>△ |rowspan=2|[[File:Regular_star_figure_2(15,7).svg|50px|{30/14}=2{15/7}]] |rowspan=2|{{sfrac|15|7}} | |colspan=2|[[W:Triacontagon#Triacontagram|{30/14}=2{15/7}]] |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2| |rowspan=2|<br>1200<br> |rowspan=2|{{blue|<big>'''4'''</big>}}<br>{3,3} |- style="background: seashell;"| |164.5~° |{{radic|3.93~}} |1.982~ |- style="background: paleturquoise;"| |rowspan=2|#15<br><small>△☐𝜙</small> |rowspan=2|[[File:Regular_star_figure_15(2,1).svg|50px|30/15}=15{2}]] |rowspan=2|2 |𝝅 |colspan=2|[[W:Diameter|diameter]] |rowspan=2| |rowspan=2|{{red|<big>'''75'''</big>}}{{Efn|name=inscribed counts}}<br>4 |rowspan=2|<br>8 |rowspan=2|<br>12 |rowspan=2|<br>48 |rowspan=2|<br>60 |rowspan=2|<br>240 |rowspan=2|<br>300{{Efn|name=rays and bases}} |rowspan=2|{{blue|<big>'''1'''</big>}}<br><br> |- style="background: paleturquoise;"| |180° |{{radic|4}} |2 |- !colspan=6|Squared lengths total{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} | style="background: seashell;"|25 | style="background: paleturquoise;"|64 | style="background: paleturquoise;"|256 | style="background: paleturquoise;"|576 | style="background: yellow;"| | style="background: yellow;"|14400 | style="background: seashell;"| | style="background: seashell;"|360000{{Efn|name=additional 120-cell chords}} !<big>{{blue|'''300'''}}</big> |} [[File:15 major chords.png|thumb|300px|The major{{Efn|name=additional 120-cell chords}} chords #1 - #15 join vertex pairs which are 1 - 15 edges apart on a Petrie polygon.{{Efn|Drawing the fan of chords with #1 and #11 at a different origin than all the others is an artistic choice, since all the chords are incident at every vertex. We could just as well have drawn all the chords from the same origin vertex, but this arrangement notices the parallel relationship between #8 and #11.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] The annotated chord table is a complete [[W:Bill of materials|bill of materials]] for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table. The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the '''#3''' chord row, the 600-cell's 72 great decagons contain 720 '''#3''' chords in all. The '''{{red|red}}''' integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of <big>{{red|'''25'''}}</big> disjoint 24-cells (25 * 24 vertices = 600 vertices). The '''{{green|green}}''' integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains <big>{{green|'''225'''}}</big> distinct 24-cells which share components. The '''{{blue|blue}}''' integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the '''#3''' chord row, <big>{{blue|'''24'''}}</big> '''#3''' chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral [[W:Vertex figure|vertex figure]] 2{3,5}. In total <big>{{blue|'''300'''}}</big> major chords{{Efn|name=additional 120-cell chords}} of 15 distinct lengths meet at each vertex of the 120-cell. === Relationships among interior polytopes === The 120-cell is the compound of all five of the other regular convex 4-polytopes.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.{{Efn|The 120-cell contains instances of all of the regular convex 1-polytopes, 2-polytopes, 3-polytopes and 4-polytopes, ''except'' for the regular polygons {7} and above, most of which do not occur. {10} is a notable exception which ''does'' occur. Various regular [[W:Skew polygon|skew polygon]]s {7} and above occur in the 120-cell, notably {11},{{Efn|name={30/11}-gram}} {15}{{Efn|name=120-cell characteristic rotation}} and {30}.{{Efn|name=two coaxial Petrie 30-gons}}|name=elements}} It is a four-dimensional [[W:Jigsaw puzzle|jigsaw puzzle]] in which all those polytopes are the parts.{{Sfn|Schleimer & Segerman|2013}} Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.{{Sfn|Stillwell|2001}} The regular 1-polytope occurs in only [[#Chords|15 distinct lengths]] in any of the component polytopes of the 120-cell.{{Efn|name=additional 120-cell chords}} By [[W:Alexandrov's uniqueness theorem|Alexandrov's uniqueness theorem]], convex polyhedra with shapes distinct from each other also have distinct [[W:Metric spaces|metric spaces]] of surface distances, so each regular 4-polytope has its own unique subset of these 15 chords. Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four {{background color|paleturquoise|[[24-cell#Hypercubic chords|hypercubic chords]]}} {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}} are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built solely from them. {{see also|W:24-cell#Relationships among interior polytopes|label 1=24-cell § Relationships among interior polytopes}} An additional 4 of the 15 chords are required to build the 600-cell. The four {{background color|yellow|[[600-cell#Golden chords|golden chords]]}} are square roots of irrational fractions that are functions of {{radic|5}}. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built solely from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra. {{see also|W:600-cell#Icosahedra|label 1=600-cell § Icosahedra}} All 15 major chords, and 15 other distinct chordal distances (the minor chords [[120-cell#Geodesic rectangles|enumerated below]]), occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are {{background color|#FFCCCC|[[5-cell#Boerdijk–Coxeter helix|regular 5-cells and {{radic|5/2}} chords]].}}{{Efn|Dodecahedra emerge as ''visible'' features in the 120-cell, but they also occur in the 600-cell as ''interior'' polytopes.{{Sfn|Coxeter|1973|p=298|loc=Table V: (iii) Sections of {3,3,5} beginning with a vertex}}}} The relationships between the ''regular'' 5-cell (the [[W:Simplex|simplex]] regular 4-polytope) and the other regular 4-polytopes are manifest directly only in the 120-cell.{{Efn|There is a geometric relationship between the regular 5-cell (4-simplex) and the regular 16-cell (4-orthoplex), but it is manifest only indirectly through the [[W:Tetrahedron|3-simplex]] and [[W:5-orthoplex|5-orthoplex]]. An [[W:simplex|<math>n</math>-simplex]] is bounded by <math>n+1</math> vertices and <math>n+1</math> (<math>n</math>-1)-simplex facets, and has <math>z+1</math> long diameters (its edges) of length <math>\sqrt{n+1}/\sqrt{n}</math> radii. An [[W:orthoplex|<math>n</math>-orthoplex]] is bounded by <math>2n</math> vertices and <math>2^n</math> (<math>n</math>-1)-simplex facets, and has <math>n</math> long diameters (its orthogonal axes) of length <math>2</math> radii. An [[W:hypercube|<math>n</math>-cube]] is bounded by <math>2^n</math> vertices and <math>2n</math> (<math>n</math>-1)-cube facets, and has <math>2^{n-1}</math> long diameters of length <math>\sqrt{n}</math> radii.{{Efn|The <math>n</math>-simplex's facets are larger than the <math>n</math>-orthoplex's facets. For <math>n=4</math>, the edge lengths of the 5-cell and 16-cell and 8-cell are in the ratio of <math>\sqrt{5}</math> to <math>\sqrt{4}</math> to <math>\sqrt{2}</math>.|name=root 5/root 4/root 2}} The <math>\sqrt{3}</math> long diameters of the 3-cube are shorter than the <math>\sqrt{4}</math> axes of the 3-orthoplex. The [[16-cell#Coordinates|coordinates of the 4-orthoplex]] are the permutations of <math>(0,0,0,\pm 1)</math>, and the 4-space coordinates of one of its 16 facets (a 3-simplex) are the permutations of <math>(0,0,0,1)</math>.{{Efn|Each 3-facet of the 4-orthoplex, a tetrahedron permuting <math>(0,0,0,1)</math>, and its completely orthogonal 3-facet permuting <math>(0,0,0,-1)</math>, comprise all 8 vertices of the 4-orthoplex. Uniquely, the 4-orthoplex is also the 4-[[W:demihypercube|demicube]], half the vertices of the 4-cube. This relationship among the 4-simplex, 4-orthoplex and 4-cube is unique to <math>n=4</math>. The 4-orthoplex's completely orthogonal 3-simplex facets are a pair of 3-demicubes which occupy alternate vertices of completely orthogonal 3-cubes in the same 4-cube. Projected orthogonally into the same 3-hyperplane, the two 3-facets would be two tetrahedra inscribed in the same 3-cube. (More generally, completely orthogonal polytopes are mirror reflections of each other.)|name=4-simplex-orthoplex-cube relation}} The <math>\sqrt{4}</math> long diameters of the 4-cube are the same length as the <math>\sqrt{4}</math> axes of the 4-orthoplex. The [[W:5-orthoplex#Cartesian coordinates|coordinates of the 5-orthoplex]] are the permutations of <math>(0,0,0,0,\pm 1)</math>, and the 5-space coordinates of one of its 32 facets (a 4-simplex) are the permutations of <math>(0,0,0,0,1)</math>.{{Efn|Each 4-facet of the 5-orthoplex, a 4-simplex (5-cell) permuting <math>(0,0,0,0,1)</math>, and its completely orthogonal 4-facet permuting <math>(0,0,0,0,-1)</math>, comprise all 10 vertices of the 5-orthoplex.}} The <math>\sqrt{5}</math> long diameters of the 5-cube are longer than the <math>\sqrt{4}</math> axes of the 5-orthoplex.|name=simplex-orthoplex-cube relation}} The 600-point 120-cell is a compound of 120 disjoint 5-point 5-cells, and it is also a compound of 5 disjoint 120-point 600-cells (two different ways). Each 5-cell has one vertex in each of 5 disjoint 600-cells, and therefore in each of 5 disjoint 24-cells, 5 disjoint 8-cells, and 5 disjoint 16-cells.{{Efn|No vertex pair of any of the 120 5-cells (no [[5-cell#Geodesics and rotations|great digon central plane of a 5-cell]]) occurs in any of the 675 16-cells (the 675 [[16-cell#Coordinates|Cartesian basis sets of 6 orthogonal central planes]]).{{Efn|name=rays and bases}}}} Each 5-cell is a ring (two different ways) joining 5 disjoint instances of each of the other regular 4-polytopes.{{Efn|name=distinct circuits of the 5-cell}} {{see also|W:5-cell#Geodesics and rotations|label 1=5-cell § Geodesics and rotations}} === Compound of five 600-cells === [[File:Great dodecagon of the 120-cell.png|thumb|300px|The 120-cell has 200 central planes that each intersect 12 vertices, forming an irregular dodecagon with alternating edges of two different lengths. Inscribed in the dodecagon are two regular great hexagons (black),{{Efn|name=great hexagon}} two irregular great hexagons ({{Color|red|red}}),{{Efn|name=irregular great hexagon}} and four equilateral great triangles (only one is shown, in {{Color|green|green}}).]] The 120-cell contains ten 600-cells which can be partitioned into five completely disjoint 600-cells two different ways.{{Efn|name=2 ways to get 5 disjoint 600-cells}} As a consequence of being a compound of five disjoint 600-cells, the 120-cell has 200 irregular great dodecagon {12} central planes, which are compounds of several of its great circle polygons that share the same central plane, as illustrated. The 200 {12} central planes originate as the compounds of the hexagonal central planes of the 25 disjoint inscribed 24-cells and the digon central planes of the 120 disjoint inscribed regular 5-cells; they contain all the 24-cell and 5-cell edges, and also the 120-cell edges. Thus the edges and characteristic rotations{{Efn|Every class of discrete isoclinic rotation{{Efn|name=isoclinic rotation}} is characterized by its rotation and isocline angles and by which set of Clifford parallel central planes are its invariant planes of rotation. The '''characteristic isoclinic rotation of a 4-polytope''' is the class of discrete isoclinic rotation in which the set of invariant rotation planes contains the 4-polytope's edges; there is a distinct left (and right) rotation for each such set of Clifford parallel central planes (each [[W:Hopf fibration|Hopf fibration]] of the edge planes). If the edges of the 4-polytope form regular great circles, the rotation angle of the characteristic rotation is simply the edge arc-angle (the edge chord is simply the rotation chord). But in a regular 4-polytope with a tetrahedral vertex figure{{Efn|name=non-planar geodesic circle}} the edges do not form regular great circles, they form irregular great circles in combination with another chord. For example, the #1 chord edges of the 120-cell are edges of an [[#Compound of five 600-cells|irregular great dodecagon]] which also has #4 chord edges. In such a 4-polytope, the rotation angle is not the edge arc-angle; in fact it is not necessarily the arc of any vertex chord.{{Efn|name=12° rotation angle}}|name=characteristic rotation}} of the regular 5-cell, the 8-cell hypercube, the 24-cell, and the 120-cell all lie in these same 200 rotation planes.{{Efn|name=edge rotation planes}} Each of the ten 600-cells occupies the entire set of 200 planes. The 120-cell's irregular [[#Other great circle constructs|dodecagon {12} great circle polygon]] has 6 short edges (#1 [[#Chords|chords]] marked {{Color|red|𝜁}}) alternating with 6 longer dodecahedron cell-diameters ({{Color|magenta|#4}} chords).{{Efn|name=dodecahedral cell metrics}} Inscribed in the irregular great dodecagon are two irregular great hexagons ({{color|red|red}}) in alternate positions.{{Efn|name=irregular great hexagon}} Two ''regular'' great hexagons with edges of a third size ({{radic|1}}, the #5 chord) are also inscribed in the dodecagon.{{Efn|name=great hexagon}} The 120-cell's irregular great dodecagon planes, its irregular great hexagon planes, its regular great hexagon planes, and its equilateral great triangle planes, are the same set of 200 dodecagon planes. They occur as 100 completely orthogonal pairs, and they are the ''same'' 200 central planes each containing a [[600-cell#Hexagons|hexagon]] that are found in ''each'' of the 10 inscribed 600-cells. There are exactly 400 regular hexagons in the 120-cell (two in each dodecagon central plane), and each of the ten 600-cells contains its own distinct subset of 200 of them (one from each dodecagon central plane). Each 600-cell contains only one of the two opposing regular hexagons inscribed in any dodecagon central plane, just as it contains only one of two opposing tetrahedra inscribed in any dodecahedral cell. Each 600-cell is disjoint from 4 other 600-cells, and shares regular hexagons with 5 other 600-cells.{{Efn|Each regular great hexagon is shared by two 24-cells in the same 600-cell,{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} and each 24-cell is shared by two 600-cells.{{Efn|name=two 600-cells share a 24-cell}} Each regular hexagon is shared by four 600-cells.|name=hexagons 24-cells and 600-cells}} Each disjoint pair of 600-cells occupies the opposing pair of disjoint regular hexagons in every dodecagon central plane. Each non-disjoint pair of 600-cells intersects in 16 hexagons that comprise a 24-cell. The 120-cell contains 9 times as many distinct 24-cells (225) as disjoint 24-cells (25).{{Efn|name=rays and bases}} Each 24-cell occurs in 9 600-cells, is absent from just one 600-cell, and is shared by two 600-cells. ===Concentric hulls=== [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|left|640px| Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an Overall Hull that is a [[W:Chamfered dodecahedron|chamfered dodecahedron]] of Norm={{radic|8}}.<br /> Hulls 1 - 8 are the 8 sections of the 120-cell beginning with a cell (Hull 1).<br /> Hulls 1, 2, & 7 are each pairs of [[W:Dodecahedron|dodecahedron]]s.<br /> Hull 3 is a pair of [[W:Icosidodecahedron|icosidodecahedron]]s.<br /> Hulls 4 & 5 are each pairs of [[W:Truncated icosahedron|truncated icosahedron]]s.<br /> Hull 6 is a pair of semi-regular [[W:Rhombicosidodecahedron|rhombicosidodecahedron]]s.<br /> Hull 8 is a single non-uniform [[W:Rhombicosidodecahedron#Names|rhombicosidodecahedron]], the central section.<br /> A more detailed visualization of these 15 simplified sections, with subgroup sections where the inscribed solid has more than one permutation in its orbit, is available [https://commons.wikimedia.org/wiki/File:Cell_First_533_120-Cell_Sections.svg here].]] {{Clear}} These hulls illustrate Coxeter's sections 1₃ - 8₃ of the 120-cell, the sections beginning with a cell (hull #1).{{Sfn|Coxeter|1973|p=299|loc=Table V (iv) Sections of {5,3,3} beginning with a cell (right half of table)}} A ''section'' is a flat 3-dimensional hyperplane slice through the [[W:3-sphere|3-sphere]]: a 2-sphere (ordinary sphere). It is dimensionally analogous to a flat 2-dimensional plane slice through a 2-sphere: a 1-sphere (ordinary circle). The hulls are illustrated as if they were all the same size, but actually they increase in radius as numbered: they are concentric 2-spheres that nest inside each other. Every cell of the 120-cell is the smallest hull in its own set of 8 concentric hulls. There are 120 distinct sets of hulls. The 120-cell actually has 15 sections beginning with a cell, numbered 1 - 15 with number 8 in the center. After increasing in size from 1 to 8, the hulls get smaller again. Sections 1 and 15 are both a hull #1, the smallest hull, a dodecahedral cell of the 120-cell. Section #8 is the central section, the largest hull, with the same radius as the 120-cell. Except for the central section #8, the sections occur in parallel pairs, on either side of the central section. Hull #8 is dimensionally analogous to the equator, while hulls #1 - #7 are dimensionally analogous to lines of latitude. There are 120 of each kind of hull #1 - #7 in the 120-cell, but only 60 of the central hull #8. {{Clear}} The 120-cell also has 30 sections beginning with a vertex, illustrated below. Like the sections beginning with a cell illustrated above, the vertex-first sections are also flat 3-dimensional hyperplane slices through the 3-sphere, and polyhedra that nest inside each other as concentric 2-spheres. Section 0₀ is the vertex itself. Section 1₀ is the 120-cell's tetrahedral vertex figure. Sections 1₀ - 29₀ are described in more detail in [[120-cell#Geodesic rectangles|§Geodesic rectangles]], below. {{Clear}} [[File:Vertex_First_533_120-Cell_Sections.svg|thumb|left|640px| Coxeter's sections 0₀ - 30₀ of the 120-cell, the sections beginning with a vertex, showing the orbit sections and subgroup sections (when the inscribed solid has more than one permutation in its orbit), as well as the convex hull of each orbit on the right.]] {{Clear}} === Geodesic rectangles === The 30 distinct chords{{Efn|name=additional 120-cell chords}} found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: {{Background color|palegreen|△ planes that intersect {12} vertices}} in an [[#Compound of five 600-cells|irregular great dodecagon]], {{Background color|yellow|<big>𝜙</big> planes that intersect {10} vertices}} in a regular decagon, and <big>☐</big> planes that intersect {4} vertices in several kinds of {{Background color|gainsboro|rectangle}}, including a {{Background color|seashell|square}}. Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges. Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing [[#Concentric hulls|polyhedral sections]] of the 120-cell beginning with a vertex (the 0₀ section), as illustrated above. The correspondence is that each 120-cell vertex is surrounded in curved 3-space <math>S_3</math> by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius in Euclidean 3-space <math>R_3</math>.{{Efn|In the curved 3-dimensional space <math>S_3</math> of the 120-cell's surface, each of the 600 vertices is surrounded by 15 pairs of polyhedral sections, each section at the "radial" distance of one of the 30 distinct chords. The vertex is not actually at the center of the polyhedron, because it is displaced in the fourth dimension out of the section's hyperplane, so that the ''apex'' vertex and its surrounding ''base'' polyhedron form a [[W:Polyhedral pyramid|polyhedral pyramid]]. The characteristic chord is radial around the apex, as the pyramid's lateral edges.}} There are 600 distinct sets of 15 hulls. The #1 chord is the radius in <math>S_3</math> of the 1₀ section, the tetrahedral vertex figure of the 120-cell.{{Efn|name=#2 chord}} The #14 chord is the radius in <math>S_3</math> of its congruent opposing 29₀ section. The #7 chord is the radius in <math>S_3</math> of the central vertex-first section of the 120-cell, in which two opposing 15₀ sections are coincident. Each vertex is surrounded by two instances of each polyhedron, at the near and far radial distances of the polyhedron's 180° complementary chords, but because curved space <math>S_3</math> begins to close back up on itself after the #7 90° chord, the near and far concentric polyhedra are the same size. Each chord length is given three ways (on successive lines): for the unit-radius 120-cell as a square root, for the unit-radius 120-cell, and for the unit-edge 120-cell.{{Efn|We give chord lengths as unit-radius square roots in these articles, even when they are integers (e.g. the long diameter is {{radic|4}}). Our usual metric is unit-radius, which reveals relationships among successive 4-polytopes,{{Efn|name=4-polytopes ordered by size and complexity}} but Coxeter{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} and Steinbach{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with "fan of chords" diagrams.|p=23|loc=Figure 3}} use unit-edge, which reveals relationships among successive chords.|name=metrics}} To the left of this last unit-edge metric, its reciprocal^-1 is given. The reciprocal is the long radius of a regular ''n''₀-polygon with unit-radius 120-cell edges (#1 chords) as its edges; but this does not imply that the section ''n''₀ polyhedron contains any ''n''₀ polygons.{{Efn|The 120-cell contains no regular {30} central polygons, although its Petrie polygon is a skew regular {30}. Therefore the edge of the regular triacontagon {30} is not a chord of the 120-cell represented in this table. Nevertheless these metrics of the {30} are relevant:<br> :Unit-radius {30}: ::Edge <small><math>E = 2 \sin{\pi/30} \approx \sqrt{0.0437} \approx 0.209</math></small> :Unit-edge {30}: ::Radius <small><math>R_{ue} = 1/E \approx 4.783</math></small> :{30} with 120-cell edges: ::Edge <small><math>\zeta \approx 0.270~</math></small> ::<small><math>E \approx 0.774 \times \zeta</math></small> ::Radius <small><math>R_\zeta \approx 1.292</math></small> |name=triacontagon metrics}} {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |#0<br><br>0₀ | |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |<math>\pi</math> |{{radic|4}} |{{radic|4}} | rowspan="3" |#15<br><br>30₀ |- style="background: palegreen;" | |0° |0 |0 |180° |2 |2 |- style="background: palegreen;" | | |0 |<small><math>0\times\zeta</math></small> |0.135~^-1 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#1<br><br>1₀ |𝞯 |{{radic|0.𝜀}}{{Efn|name=fractional square roots}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} | |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |#14<br><br>29₀ |- style="background: palegreen;" | |15.5~°{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}} |0.270~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |164.5~° |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |1^-1 |1 |<small><math>1\times\zeta</math></small> |0.136~^-1 |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#2<br><br>2₀ |{{Efn|name=#2 chord}} |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 | |{{radic|3.81~}} | | rowspan="3" |#13<br><br>28₀ |- style="background: gainsboro;" | |25.2~° |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |154.8~° |1.952~ | |- style="background: gainsboro;" | |0.618~^-1 |1.618~ |<small><math>\phi\times\zeta</math></small> |0.138~^-1 |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |#3<br><br>3₀ |<math>\pi / 5</math> |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |<math>4\pi / 5</math> |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |#12<br><br>27₀ |- style="background: yellow;" | |36° |0.618~ |<small><math>1 / \phi</math></small> |144°{{Efn|name=dihedral}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.437~^-1 |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |0.142~^-1 |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |#3⁺<br><br>4₀ | |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |#12⁻<br><br>26₀ |- style="background: gainsboro;" | |41.4~° |0.707~ |<small><math>\sqrt{2}/2</math></small> |138.6~° |1.871~ | |- style="background: gainsboro;" | |0.382~^-1 |2.618~ |<small><math>\phi^2\times\zeta</math></small> |0.144~^-1 |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |#4<br><br>5₀ | |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons{{Efn|This illustration shows just one of three related irregular great dodecagons that lie in three distinct △ central planes. Two of them (not shown) lie in Clifford parallel (disjoint) dodecagon planes, and share no vertices. The {{Color|blue}} central rectangle of #4 and #11 edges lies in a third dodecagon plane, not Clifford parallel to either of the two disjoint dodecagon planes and intersecting them both; it shares two vertices (a {{radic|4}} axis of the rectangle) with each of them. Each dodecagon plane contains two irregular great hexagons in alternate positions (not shown). Thus each #4 chord of the great rectangle shown is a bridge between two Clifford parallel irregular great hexagons that lie in the two dodecagon planes which are not shown.{{Efn|Isoclinic rotations take Clifford parallel planes to each other, as planes of rotation tilt sideways like coins flipping.{{Efn|name=isoclinic rotation}} The #4 chord{{Efn|name=#4 isocline chord}} bridge is significant in an isoclinic rotation in ''regular'' great hexagons (the [[600-cell#Hexagons|24-cell's characteristic rotation]]), in which the invariant rotation planes are a subset of the same 200 dodecagon central planes as the 120-cell's characteristic rotation (in ''irregular'' great hexagons).{{Efn|name=120-cell characteristic rotation}} In each 12° arc{{Efn|name=120-cell rotation angle}} of the 24-cell's characteristic rotation of the 120-cell, every ''regular'' great hexagon vertex is displaced to another vertex, in a Clifford parallel regular great hexagon that is a #4 chord away. Adjacent Clifford parallel regular great hexagons have six pairs of corresponding vertices joined by #4 chords. The six #4 chords are edges of six distinct great rectangles in six disjoint dodecagon central planes which are mutually Clifford parallel.|name=#4 isocline chord bridge}}|name=dodecagon rotation}}<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" |{{Efn|name=#4 isocline chord bridge}} | |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |#11<br><br>25₀ |- style="background: palegreen;" | |44.5~° |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |135.5~° |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.357~^-1 |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |0.146~^-1 |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#4⁺<br><br>6₀ | |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |#11⁻<br><br>24₀ |- style="background: gainsboro;" | |49.1~° |0.831~ | |130.9~° |1.819~ | |- style="background: gainsboro;" | |0.325~^-1 |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |0.148~^-1 |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁻<br><br>7₀ | |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | | |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |#10⁺<br><br>23₀ |- style="background: gainsboro;" | |56° |0.939~ | |124° |1.766~ | |- style="background: gainsboro;" | |0.288~^-1 |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |0.153~^-1 |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |#5<br><br>8₀ |<math>\pi / 3</math> |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]{{Efn|name=great hexagon}}<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |<small><math>2\pi / 3</math></small> |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |#10<br><br>22₀ |- style="background: palegreen;" | |60° |1 | |120° |1.732~ | |- style="background: palegreen;" | |0.270~^-1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |0.156~^-1 |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#5⁺<br><br>9₀ | |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |#10⁻<br><br>21₀ |- style="background: gainsboro;" | |66.1~° |1.091~ | |113.9~° |1.676~ | |- style="background: gainsboro;" | |0.247~^-1 |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |0.161~^-1 |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁻<br><br>10₀ | |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |#9⁺<br><br>20₀ |- style="background: gainsboro;" | |69.8~° |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |110.2~° |1.640~ | |- style="background: gainsboro;" | |0.236~^-1 |4.236~ |<small><math>\phi^3\times\zeta</math></small> |0.165~^-1 |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |#6<br><br>11₀ |<math>2\pi/5</math> |{{radic|1.𝚫}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]{{Efn|name=great pentagon}}<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |<math>3\pi / 5</math> |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |#9<br><br>19₀ |- style="background: yellow;" | |72° |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |108° |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |0.230~^-1 |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |0.167~^-1 |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |#6⁺⁻<br><br>12₀ | |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]{{Efn|The [[5-cell#Geodesics and rotations|regular 5-cell has only digon central planes]] intersecting two vertices. The 120-cell with 120 inscribed regular 5-cells contains great rectangles whose longer edges are these digons, the edges of inscribed 5-cells of length {{radic|2.5}}. Three disjoint rectangles occur in one {12} central plane, where the six #8 {{radic|2.5}} chords belong to six disjoint 5-cells. The 12₀ sections and 18₀ sections are regular tetrahedra of edge length {{radic|2.5}}, the cells of regular 5-cells. The regular 5-cells' ten triangle faces lie in those sections; each of a face's three {{radic|2.5}} edges lies in a different {12} central plane.|name=5-cell rotation}}<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 | |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |#8<br><br>18₀ |- style="background: palegreen;" | |75.5~° |1.224~ | |104.5~° |1.581~ | |- style="background: palegreen;" | |0.221~^-1 |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |0.171~^-1 |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺<br><br>13₀ | |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.31~}} | | rowspan="3" |#8⁻<br><br>17₀ |- style="background: gainsboro;" | |81.1~° |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |98.9~° |1.520~ | |- style="background: gainsboro;" | |0.208~⁻¹ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |0.178~^-1 |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |#6⁺⁺<br><br>14₀ | |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | | |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |#7⁺<br><br>16₀ |- style="background: gainsboro;" | |84.5~° |1.345~ | |95.5~° |1.480~ | |- style="background: gainsboro;" | |0.201~⁻¹ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |0.182~^-1 |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |#7<br><br>15₀ |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]{{Efn|name=rays and bases}}<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |<math>\pi / 2</math> |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |#7<br><br>15₀ |- style="background: seashell;" | |90° |1.414~ | |90° |1.414~ | |- style="background: seashell;" | |0.191~⁻¹ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |0.191~^-1 |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotation{{Efn|name=isoclinic rotation}} of a distinct class,{{Efn|name=characteristic rotation}} which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.{{Efn|In the 120-cell, completely orthogonal to every great circle polygon lies another great circle polygon of the same kind. The set of Clifford parallel invariant planes of a distinct isoclinic rotation is a set of such completely orthogonal pairs.{{Efn|name=Clifford parallel invariant planes}}}} There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.{{Efn|Each kind of rotation plane has its characteristic fibration divisor, denoting the number of fiber bundles of Clifford parallel great circle polygons (of each distinct kind) that are found in rotation planes of that kind. Each bundle covers all the vertices of the 120-cell exactly once, so the total number of vertices in the great circle polygons of one kind, divided by the number of bundles, is always 600, the number of distinct vertices. For example, "400 irregular great hexagons" / 4.}} In each class of rotation,{{Efn|[[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]] are defined by at least one pair of completely orthogonal{{Efn|name=perpendicular and parallel}} central planes of rotation which are ''invariant'', which means that all points in the plane stay in the plane as the plane moves. A distinct left (and right) isoclinic{{Efn|name=isoclinic}} rotation may have multiple pairs of completely orthogonal invariant planes, and all those invariant planes are mutually [[W:Clifford parallel|Clifford parallel]]. A distinct class of discrete isoclinic rotation has a characteristic kind of great polygon in its invariant planes.{{Efn|name=characteristic rotation}} It has multiple distinct left (and right) rotation instances called ''fibrations'', which have disjoint sets of invariant rotation planes. The fibrations are disjoint bundles of Clifford parallel circular ''fibers'', the great circle polygons in their invariant planes.|name=Clifford parallel invariant planes}} vertices rotate on a distinct kind of circular geodesic isocline{{Efn|name=isocline}} which has a characteristic circumference, skew Clifford polygram{{Efn|name=Clifford polygon}} and chord number, listed in the Rotation column above.{{Efn|The 120-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into the sets of Clifford parallel invariant rotation planes of 25 distinct classes of (double) rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} ===Polyhedral graph=== Considering the [[W:Adjacency matrix|adjacency matrix]] of the vertices representing the polyhedral graph of the unit-radius 120-cell, the [[W:Graph diameter|graph diameter]] is 15, connecting each vertex to its coordinate-negation at a [[W:Euclidean distance|Euclidean distance]] of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582. The vertices of the 120-cell polyhedral graph are [[W:Vertex coloring|3-colorable]]. The graph is [[W:Eulerian path|Eulerian]] having degree 4 in every vertex. Its edge set can be decomposed into two [[W:Hamiltonian path|Hamiltonian cycles]].<ref>{{cite book| author = Carlo H. Séquin | title = Symmetrical Hamiltonian manifolds on regular 3D and 4D polytopes | date = July 2005 | pages = 463–472 | publisher = Mathartfun.com | isbn = 9780966520163 | url = https://archive.bridgesmathart.org/2005/bridges2005-463.html#gsc.tab=0 | access-date=March 13, 2023}}</ref> === Constructions === The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}} It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the [[600-cell]],{{Efn|name=2 ways to get 5 disjoint 600-cells}} just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the [[24-cell|24-cell]],{{Efn|In the 120-cell, each 24-cell belongs to two different 600-cells.{{Sfn|van Ittersum|2020|p=435|loc=§4.3.5 The two 600-cells circumscribing a 24-cell}} The 120-cell contains 225 distinct 24-cells and can be partitioned into 25 disjoint 24-cells, so it is the convex hull of a compound of 25 24-cells.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=5|loc=§2 The Labeling of H4}}|name=two 600-cells share a 24-cell}} the 24-cell can be deconstructed into three distinct instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.{{Efn|The 120-cell has 600 vertices distributed symmetrically on the surface of a 3-sphere in four-dimensional Euclidean space. The vertices come in antipodal pairs, and the lines through antipodal pairs of vertices define the 300 '''rays''' [or axes] of the 120-cell. We will term any set of four mutually orthogonal rays (or directions) a '''[[W:Orthonormal basis|basis]]'''. The 300 rays form 675 bases, with each ray occurring in 9 bases and being orthogonal to its 27 distinct companions in these bases and to no other rays. The rays and bases constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 300₉675₄ to indicate that each ray belongs to 9 bases, and each basis contains 4 rays.{{Sfn|Waegell|Aravind|2014|loc=§2 Geometry of the 120-cell: rays and bases|pp=3-4}} Each basis corresponds to a distinct [[16-cell#Coordinates|16-cell]] containing four orthogonal axes and six orthogonal great squares. 75 completely disjoint 16-cells containing all 600 vertices of the 120-cell can be selected from the 675 distinct 16-cells.{{Efn|name=rotated 4-simplexes are completely disjoint}}|name=rays and bases}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse [[W:Golden ratio|golden ratio]]), but the 120-cell's edge length is ~0.270 times its radius. The 120-cell is also the convex hull of the regular compound of 120 disjoint regular 5-cells. This can be seen to be equivalent to the compound of 5 disjoint 600-cells, as follows. Beginning with a single 120-point 600-cell, expand each vertex into a regular 5-cell. For each of the 120 vertices, add 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point 600-cells: a 120-cell. ==== Dual 600-cells ==== [[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|right|Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.]] Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius ({{sfrac|φ²|{{radic|8}}}} ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ²{{radic|2}} ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The [[#Unit radius coordinates|unit radius 120-cell]] (with edge-length {{sfrac|1|φ²{{radic|2}}}} ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius {{sfrac|{{radic|8}}|φ²}} ≈ 1.080. [[File:Dodecahedron_vertices.svg|thumb|150px|right|One of the five distinct cubes inscribed in the dodecahedron (dashed lines). Two opposing tetrahedra (not shown) lie inscribed in each cube, so ten distinct tetrahedra (one from each 600-cell in the 120-cell) are inscribed in the dodecahedron.{{Efn|In the [[W:120-cell#Dual 600-cells|dodecahedral cell]] of the unit-radius 120-cell, the edge is the '''15.5° #1 [[#Chords|chord]]''' of the 120-cell of length <small><math>\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270</math></small>. Eight {{Color|orange}} vertices lie at the Cartesian coordinates <small><math>(\pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8}, \pm\phi^3\sqrt{8})</math></small> relative to origin at the cell center. They form a cube (dashed lines) whose edges are the '''25.2° #2 chord''' of length <small><math>\tfrac{1}{\phi\sqrt{2}} \approx 0.437</math></small> (the pentagon diagonal). The face diagonals of the cube (not drawn) are the '''36° #3 chord''' of length <small><math>\tfrac{1}{\phi} \approx 0.618</math></small> (the edges of two 600-cell tetrahedron cells inscribed in the cube). The next largest '''41.1° chord''' has length <small><math>\tfrac{1}{\sqrt{2}} \approx 0.707</math></small>. The diameter of the dodecahedron is the '''44.5° #4 chord''' of length <small><math>\tfrac{\sqrt{3}}{\phi\sqrt{2}} \approx 0.757</math></small> (the cube diagonal). If the #4 diameter is extended outside the dodecahedron in a straight line in the curved space of the 3-sphere, it is colinear with a #1 edge belonging to three neighboring dodecahedron cells, and the combined '''60° #5 chord''' has length <small><math>\sqrt{1}</math></small> (an edge of an inscribed 24-cell). If this 60° combined #4 plus #1 geodesic is further extended in a straight line by another #4 chord (the diameter of a further cell), the combined '''104.5° #8 chord''' has length <small><math>\tfrac{\sqrt{5}}{\sqrt{2}} \approx 1.581</math></small> (an edge of an inscribed regular 5-cell).|name=dodecahedral cell metrics}}]] Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius {{sfrac|φ²|{{radic|8}}}} ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given [[#√8 radius coordinates|above]] of long radius {{Radic|8}} = 2{{Radic|2}} ≈ 2.828 and edge-length {{sfrac|2|φ²}} = 3−{{radic|5}} ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ², which is smaller than {{Radic|8}} in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ²{{Radic|8}} ≈ 7.405 given by Coxeter{{Sfn|Coxeter|1973|loc=Table I(ii); "120-cell"|pp=292-293}} can be constructed in this manner just outside a 600-cell of long radius φ⁴ and edge-length φ³. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps. ==== Cell rotations of inscribed duals ==== Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells. The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.{{Sfn|Sullivan|1991|loc=The Dodecahedron|pp=4-5}} Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} This shows that the 120-cell contains, among its many interior features, 120 [[W:Compound of ten tetrahedra|compounds of ten tetrahedra]], each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.{{Efn|The 600 vertices of the 120-cell can be partitioned into those of 5 disjoint inscribed 120-vertex 600-cells in two different ways.{{Sfn|Waegell|Aravind|2014|pp=5-6}} The geometry of this 4D partitioning is dimensionally analogous to the 3D partitioning of the 20 vertices of the dodecahedron into 5 disjoint inscribed tetrahedra, which can also be done in two different ways because [[#Cell rotations of inscribed duals|each dodecahedral cell contains two opposing sets of 5 disjoint inscribed tetrahedral cells]]. The 120-cell can be partitioned in a manner analogous to the dodecahedron because each of its dodecahedral cells contains one tetrahedral cell from each of the 10 inscribed 600-cells.|name=2 ways to get 5 disjoint 600-cells}} All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.{{Efn|The 10 tetrahedra in each dodecahedron overlap; but the 600 tetrahedra in each 600-cell do not, so each of the 10 must belong to a different 600-cell.}} Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. ==== Augmentation ==== Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing [[W:Hyperpyramid|4-pyramid]]s of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a [[W:Regular dodecahedron#Cartesian coordinates|dodecahedron]].{{Efn|name=truncated apex}} Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a [[600-cell#Icosahedra|cluster of five tetrahedra]], with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.{{Efn|As we saw in the [[600-cell#Cell clusters|600-cell]], these 12 tetrahedra belong (in pairs) to the 6 [[600-cell#Icosahedra|icosahedral clusters]] of twenty tetrahedral cells which surround each cluster of five tetrahedral cells.}} The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. ==== Weyl orbits ==== Another construction method uses [[W:Quaternion|quaternion]]s and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe <math>T</math> and <math>T'</math> [[24-cell|24-cell]]s as quaternion orbit weights of D4 under the Weyl group W(D4):<br/> O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}<br/> O(1000) : V1<br/> O(0010) : V2<br/> O(0001) : V3 <math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math> With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the [[600-cell]] and the 120-cell of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the [[600-cell]] <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> * the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math> * the [[W:Dual snub 24-cell|dual snub 24-cell]] = <math>T \oplus T' \oplus S'</math>. === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]] represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations}}{{Sfn|Coxeter|1991|p=117}} <math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|3|node}} ! [[W:K-face|''k''-face]]||f_k||f₀ || f₁||f₂||f₃||[[W:vertex figure|''k''-fig]] !Notes |- align=right |A₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|3|node}} ||( ) !f₀ || 600 || 4 || 6 || 4 ||[[W:Regular tetrahedron|{3,3}]] || H₄/A₃ = 14400/24 = 600 |- align=right |A₁A₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|3|node}} ||{ } !f₁ || 2 || 1200 || 3 || 3 || [[W:Equilateral triangle|{3}]] || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |H₂A₁ ||{{Coxeter–Dynkin diagram|node_1|5|node|2|node_x|2|node}} ||[[W:Pentagon|{5}]] !f₂ || 5 || 5 || 720 || 2 || { } || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |H₃ ||{{Coxeter–Dynkin diagram|node_1|5|node|3|node|2|node_x}} ||[[W:Regular dodecahedron|{5,3}]] !f₃ || 20 || 30 || 12 ||120|| ( ) || H₄/H₃ = 14400/120 = 120 |} == Visualization == The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[24-cell|24-cell]]). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.{{Sfn|Sullivan|1991|p=15|loc=Other Properties of the 120-cell}} === Layered stereographic projection === The cell locations lend themselves to a hyperspherical description.{{Sfn|Schleimer & Segerman|2013|p=16|loc=§6.1. Layers of dodecahedra}} Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12). Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an [[W:Icosidodecahedron|icosidodecahedron]], with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles. {| class="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="4" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 12 cells | First layer of meridional cells / "[[W:Arctic Circle|Arctic Circle]]" | style="text-align: center" | 36° |- | style="text-align: center" | 3 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 60° |- | style="text-align: center" | 4 | style="text-align: center" | 12 cells | Second layer of meridional cells / "[[W:Tropic of Cancer|Tropic of Cancer]]" | style="text-align: center" | 72° |- | style="text-align: center" | 5 | style="text-align: center" | 30 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" | Equator |- | style="text-align: center" | 6 | style="text-align: center" | 12 cells | Third layer of meridional cells / "[[W:Tropic of Capricorn|Tropic of Capricorn]]" | style="text-align: center" | 108° | rowspan="4" | Southern Hemisphere |- | style="text-align: center" | 7 | style="text-align: center" | 20 cells | Non-meridian / interstitial | style="text-align: center" | 120° |- | style="text-align: center" | 8 | style="text-align: center" | 12 cells | Fourth layer of meridional cells / "[[W:Antarctic Circle|Antarctic Circle]]" | style="text-align: center" | 144° |- | style="text-align: center" | 9 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 120 cells ! colspan="3" | |} The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell. === Intertwining rings === [[Image:120-cell rings.jpg|right|thumb|300px|Two intertwining rings of the 120-cell.]] [[File:120-cell_two_orthogonal_rings.png|thumb|300px|Two orthogonal rings in a cell-centered projection]] The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized [[W:Hopf fibration|Hopf fibration]].{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}{{Sfn|Schleimer & Segerman|2013|pp=16-18|loc=§6.2. Rings of dodecahedra}}{{Sfn|Banchoff|2013}}{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}}{{Sfn|Sullivan|1991|loc=Other Properties of the 120-cell|p=15}} Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical [[W:Torsion of a curve|torsion]]. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two ([[W:Clifford torus|Clifford]]) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.{{Sfn|Zamboj|2021|loc=§5 Hopf tori corresponding to circles on B²|pp=23-29}} Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint ([[W:Clifford parallel|Clifford parallel]]) great circles. === Other great circle constructs === There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter [[#Chords|chords]], forming an [[#Compound of five 600-cells|irregular dodecagon in a central plane]]. Both these great circle paths have dual [[600-cell#Union of two tori|great circle paths in the 600-cell]]. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a [[600-cell#Decagons|decagon]].{{Efn|name=two coaxial Petrie 30-gons}} The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six [[W:Triangular bipyramids|triangular bipyramids]]) in the 600-cell. This latter path corresponds to a [[600-cell#Icosahedra|ring of six icosahedra]] meeting face to face in the [[W:Snub 24-cell|snub 24-cell]] (or [[W:Icosahedral pyramid|icosahedral pyramids]] in the 600-cell), forming a [[600-cell#Hexagons|hexagon]]. Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges [[#Chords|illustrated above]].{{Efn|name=irregular great hexagon}} Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each [[#Compound of five 600-cells|irregular great dodecagon]], in alternate positions. === 2D Orthogonal projections === [[W:Orthographic projection|Orthogonal projection]]s of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by [[W:B. L. Chilton|B. L. Chilton]].{{Sfn|Chilton|1964}} The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:Van Oss polygon|van Oss polygon]]. {| class="wikitable" |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:120-cell graph H4.svg|240px]]<br>[30]<br>(Red=1) |[[File:120-cell t0 p20.svg|240px]]<br>[20]<br>(Red=1) |[[File:120-cell t0 F4.svg|240px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:120-cell t0 H3.svg|240px]]<br>[10]<br>(Red=5, orange=10) |[[File:120-cell t0 A2.svg|240px]]<br>[6]<br>(Red=1, orange=3, yellow=6, lime=9, green=12) |[[File:120-cell t0 A3.svg|240px]]<br>[4]<br>(Red=1, orange=2, yellow=4, lime=6, green=8) |} === 3D Perspective projections === These projections use [[W:Perspective projection|perspective projection]], from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint. A comparison of perspective projections of the 3D dodecahedron to 2D (above left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. [[W:Schlegel diagram|Schlegel diagram]]s use [[W:Perspective (graphical)|perspective]] to show depth in the dimension which has been flattened, choosing a view point ''above'' a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. [[W:Stereographic projection|Stereographic projection]]s use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a [[W:3-sphere|3-sphere]]. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating [[120-cell#Animations|animations]] below, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must). {| class="wikitable" style="width:540px;" |+Comparison with regular dodecahedron |- !width=80|Projection ![[W:Dodecahedron|Dodecahedron]] !120-cell |- ![[W:Schlegel diagram|Schlegel diagram]] |align=center|[[Image:Dodecahedron schlegel.svg|220px]]<br>12 pentagon faces in the plane |align=center|[[File:Schlegel wireframe 120-cell.png|220px]]<br>120 dodecahedral cells in 3-space |- ![[W:Stereographic projection|Stereographic projection]] |align=center|[[Image:Dodecahedron stereographic projection.png|220px]] |align=center|[[Image:Stereographic polytope 120cell faces.png|220px]]<br>With transparent faces |} {|class="wikitable" |- !colspan=2|Enhanced perspective projections |- |align=center|[[Image:120-cell perspective-cell-first-02.png|240px]] |Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: * Nearest dodecahedron to the 4D viewpoint rendered in yellow * The 12 dodecahedra immediately adjoining it rendered in cyan; * The remaining dodecahedra rendered in green; * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image. |- |align=center|[[Image:120-cell perspective-vertex-first-02.png|240px]] |Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: * Four cells surrounding nearest vertex shown in 4 colors * Nearest vertex shown in white (center of image where 4 cells meet) * Remaining cells shown in transparent green * Cells facing away from 4D viewpoint culled for clarity |} === Animations === {|class="wikitable" !colspan=2|Projections to 3D of a 4D 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]] |- |align=center|[[File:120-cell.gif|256px]] |align=center|[[File:120-cell-inner.gif|256px]] |- |From outside the [[W:3-sphere|3-sphere]] in 4-space. |Inside the [[600-cell#Boundary envelopes|3D surface]] of the 3-sphere. |} In all the above projections of the 120-cell, only the edges of the 120-cell appear. All the other [[#Chords|chords]] are not shown. 600 chords converge at ''each'' of the 600 vertices. The complex [[#Relationships among interior polytopes|interior parts]] of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.{{Efn|[[File:Omnitruncated_120-cell_Coxeter_sections-subsections_projected_from_4D.svg|thumb|A full display of each section's orbits along with sub-section orbits in the 14400-point omnitruncated 120-cell.]]The 120-cell has <small><math>600^2 = 360,000</math></small> distinct chords. With all of its chords ''and their intersections'' it is the 14400 vertex [[W:Omnitruncation|omnitruncated]] 120-cell, which is identical to the omnitruncated 600-cell given the symmetry of their Coxeter-Dynkin diagrams.}} The following animation is an exception which does show some interior chords, although it does not reveal the inscribed 4-polytopes. {| class=wikitable width=540 !colspan=1|Coxeter section views |- |align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]<br>Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls. |} == Related polyhedra and honeycombs== === H₄ polytopes === The 120-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} {{H4_family}} === {p,3,3} polytopes === The 120-cell is similar to three [[W:Regular 4-polytope|regular 4-polytopes]]: the [[5-cell|5-cell]] {3,3,3} and [[W:Tesseract|tesseract]] {4,3,3} of Euclidean 4-space, and the [[W:Hexagonal tiling honeycomb|hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space. All of these have a [[W:Tetrahedral|tetrahedral]] [[W:Vertex figure|vertex figure]] {3,3}: {{Tetrahedral vertex figure tessellations small}} === {5,3,p} polytopes === The 120-cell is a part of a sequence of 4-polytopes and honeycombs with [[W:Dodecahedral|dodecahedral]] cells: {{Dodecahedral_tessellations_small}} === Tetrahedrally diminished 120-cell === Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.{{Efn|The diminishment of the 600-point 120-cell to a 480-point 4-polytope by removal of one if its 600-cells is analogous to the [[600-cell#Diminished 600-cells|diminishment of the 120-point 600-cell]] by removal of one of its 5 disjoint inscribed 24-cells, creating the 96-point [[W:Snub 24-cell|snub 24-cell]]. Similarly, the 8-cell tesseract can be seen as a 16-point [[24-cell#Diminishings|diminished 24-cell]] from which one 8-point 16-cell has been removed.}} [[File:Tetrahedrally_diminished_regular_dodecahedron.png|thumb|In the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]], 4 vertices are truncated to equilateral triangles. The 12 pentagon faces lose a vertex, becoming trapezoids.]] Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the [[W:Tetrahedrally diminished dodecahedron|tetrahedrally diminished dodecahedron]] because the 4 vertices removed formed a [[#Dual 600-cells|tetrahedron inscribed in the dodecahedron]]. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.{{Efn|name=face pentagon chord}} The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron,{{Efn|Each 120-cell vertex figure is actually a low tetrahedral pyramid, an irregular [[5-cell|5-cell]] with a regular tetrahedron base.|name=truncated apex}} each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges. The '''480-point diminished 120-cell''' may be called the '''tetrahedrally diminished 120-cell''' because its cells are tetrahedrally diminished, or the '''600-cell diminished 120-cell''' because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the '''regular 5-cells diminished 120-cell''' because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.{{Efn|name=inscribed 5-cells}} === Davis 120-cell manifold === The '''Davis 120-cell manifold''', introduced by {{harvtxt|Davis|1985}}, is a compact 4-dimensional [[W:Hyperbolic manifold|hyperbolic manifold]] obtained by identifying opposite faces of the 120-cell, whose universal cover gives the [[W:List of regular polytopes#Tessellations of hyperbolic 4-space|regular honeycomb]] [[W:order-5 120-cell honeycomb|{5,3,3,5}]] of 4-dimensional hyperbolic space. ==See also== *[[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] *[[W:57-cell|57-cell]] – an abstract regular 4-polytope constructed from 57 [[W:Hemi-dodecahedron|hemi-dodecahedra]]. *[[600-cell]] - the dual [[W:4-polytope|4-polytope]] to the 120-cell ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} * {{Citation | last1=Davis | first1=Michael W. | title=A hyperbolic 4-manifold | doi=10.2307/2044771 | year=1985 | journal=[[W:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=93 | issue=2 | pages=325–328| jstor=2044771 }} *[http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} * {{Cite journal|last1=Schleimer|first1=Saul|last2=Segerman|first2=Henry|date=2013|title=Puzzling the 120-cell|journal=Notices Amer. Math. Soc.|volume=62|issue=11|pages=1309–1316|doi=10.1090/noti1297 |arxiv=1310.3549 |s2cid=117636740|ref={{SfnRef|Schleimer & Segerman|2013}}}} {{Refend}} ==External links== * [https://www.youtube.com/watch?v=MFXRRW9goTs/ YouTube animation of the construction of the 120-cell] Gian Marco Todesco. * [http://www.theory.org/geotopo/120-cell/ Construction of the Hyper-Dodecahedron] * [http://www.gravitation3d.com/120cell/ 120-cell explorer] &ndash; A free interactive program (requires Microsoft .Net framework) that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL. [[Category:Geometry]] [[Category:Polyscheme]] k9sahbaosnak5ght2xgfrd8mvvps4nx 14 273009 2806700 2802766 2026-04-26T19:40:12Z DavidMCEddy 218607 /* Table of episodes */ typo 2806700 wikitext text/x-wiki A number of seemingly credible sources are describing an increase in political polarization worldwide. [[w:Maria Ressa|Maria Ressa]] describes how [[w:Rodrigo Duterte|Rodrigo Duterte]], former President of the [[w:Philippines|Philippines]] "started ... with five hundred volunteers<ref>Ressa (2022, pp. 147-8).</ref> (1) creating “sock puppets,” or fake accounts that attack or praise; (2) “mass reporting,” or organizing to negatively impact a targeted account; and (3) “astroturfing,” or fake posts or lies designed to look like grassroots support or interest.<ref>Ressa (2022, pp. 152-3).</ref> These actions tricked the algorithms of social media companies like Facebook and Twitter into amplifying fraudulent messages including incitements to violence and criminal prosecutions based on trumped up charges. The results easily overwhelmed honest media. [[w:Leila de Lima|Leila de Lima]], a Senator and former Secretary of Justice of the Philippines, spent years in pretrial detention before the charges were dropped for lack of evidence.<ref>Ressa (2022, p. 158ff) and Wikipedia, "[[w:Leila de Lima|Leila de Lima]]", accessed 2024-07-22.</ref> Ressa's news organization, [[w:Rappler|Rappler]].com, was ordered to close. Ressa herself was convicted on questionable charges. Both continued operating while the legal procedures against them were appealed.<ref>Ressa (2022, pp. 152-3) and Wikipedia, "[[w:Maria Ressa|Maria Ressa]]", accessed 2024-07-22.</ref> Ressa says similar procedures are making major contributions to the rise of fascism and far-right nationalist populists in the US, Europe, Turkey, India, Russia, and elsewhere.<ref>Ressa (2022, pp. 152-3).</ref> [[w:H. R. McMaster|H. R. McMaster]], former President Trump's second National Security advisor, said that "The internet and social media thus provided [Russia] with a low-cost, easy way to divide and weaken America from within."<ref>McMaster (2020, pp. 47-48).</ref> The [[w:2021 Facebook leak|2021 Facebook leak]] documented how executives of [[w:Facebook|Facebook]] and [[w:Meta Platforms|Meta]] knowingly prioritized profits over action to limit incitements to violence, even facilitating the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise would have reduced their profits. This "Category:Media reform to improve democracy" include videos of experts and activists working this issue along with 29:00 mm:ss audio files submitted to a ''Media & Democracy'' series syndicated on the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]<ref><!--Media & Democracy on Audioport-->{{cite Q|Q127839818}}</ref> plus text and space for moderated discussions. Some of this work is cited in the book on ''[[Media Literacy and You]]'', which is being written -- [[w:Crowdsourcing|crowdsourced]] -- to help humans better understand how they can counter the trend toward increasing political polarization and violence by talking politics, calmly, with respect and humility, with others with whom they may vehemently disagree, because the alternative is killing humans over misunderstanding. The goal is ''not'' to convince anyone that they are wrong. Rather it is to build relationships where humans can agree to disagree agreeably and collaborate to improve issues of common concern. == Table of episodes == {| class="wikitable sortable" |+ Episodes of "Media & Democracy" for the [[w:List of Pacifica Radio stations and affiliates|Pacifica Radio Network]] |- ! !! colspan=3 | Date !! |- ! no. || recorded !! broadcasted on [[w:KKFI|KKFI]] !! released to Pacifica !! Episode |- | 50 || 2026-04-09 || 2026-04-28 || 2026-05-02 || [[How US media threaten the health of all]] |- | 49 || 2026-04-06 || 2026-04-14 || 2026-04-18 || [[News suppressed for those who control money for the media]] |- | 48 || 2026-03-27 || 2026-03-31 || 2026-04-04 || [[Media and war]] |- | 47 || 2026-03-12 || 2026-03-17 || 2026-03-21 || [[Media literacy to dispel myths and improve public policy]] |- | 46 || 2026-02-26 || 2026-03-03 || 2026-03-07 || [[Concerns about media, especially in Germany]] |- | 45 || 2026-02-12 || 2026-02-17 || 2026-02-21 || [[Underserved serve themselves with low-power FM]] |- | 44 || 2026-01-30 || 2026-02-03 || 2026-02-07 || [[Conservative media are different]] |- | 43 || 2026-01-15 || 2026-01-20 || 2026-01-24 || [[Medill says you can help yourself by helping improve local media]] |- | 42 || 2026-01-03 || 2026-01-06 || 2026-01-10 || [[Lisa Loving on media literacy and how you can report for your community]] |- | 41 || 2015-12-11 || 2025-12-23 || 2025-12-27 || [[John Maxwell Hamilton on American propaganda]] |- | 40 || 2025-12-05 || 2025-12-09 || 2025-12-13 || [[You can better protect yourself from Big Tech]] |- | 39 || 2025-11-20 || 2025-11-25 || 2025-11-29 || [[Differences between media outlets including coverage of Gaza]] |- | 38 || 2025-11-06 || 2025-11-11 || 2025-11-15 || [[Media & Democracy lessons for the future]] |- | 37 || 2025-10-23 || 2025-20-28 || 2025-11-01 || [[Media reform initiatives in West Africa]] |- | 36 || 2025-10-03 || 2025-10-14 || 2025-10-18 || [[Seth Radwell says that the two Enlightenments tell us how to heal US political polarization]] |- | 35 || 2025-09-25 || 2025-09-30 || 2025-10-04 || [[Media Reform Coalition challenges anti-democratic media bias in the UK]] |- | 34 || 2025-09-12 || 2025-09-16 || 2025-09-20 || [[Fighting back against the campaign of censorship and control]] |- | 33 || 2025-08-28 || 2025-09-02 || 2025-08-06 || [[The role of the media in conflict]] |- | 32 || 2025-07-31 || 2025-08-19 || 2025-08-21 || [[Evidence-informed public policy]] |- | 31 || 2025-08-01 || 2025-08-05 || 2025-08-09 || [[What the Left can learn from Fox]] |- | 30 || 2025-07-17 || 2025-07-22 || 2025-07-26 || [[Democratic delusions: Fix the media to fix democracy]] |- | 29 || 2025-07-03 || 2025-07-08 || 2025-07-12 || [[News from Germany 1900-1945 and implications for today]] |- | 28 || 2025-06-12 || 2025-06-24 || 2025-06-28 || [[How news impacts democracy per USD Communications Professor Nik Usher]] |- | 27 || 2025-06-08 || 2025-06-10 || 2025-06-14 || [[Media concentration per Columbia History Professor Richard John]] |- | 26 || 2025-05-21 || 2025-05-27 || 2025-05-31 || [[Dean Starkman and the watchdog that didn't bark]] |- | 25 || 2025-05-08 || 2025-05-13 || 2025-05-17 || [[Freedom of the Press Foundation says...]] |- | 24 || 2025-04-24 || 2025-04-29 || 2025-05-03 || [[Canadian journalist Marc Edge on media reform to improve democracy]] |- | 23 || 2025-04-10 || 2025-04-15 || 2025-04-19 || [[The value of indigenous and community radio]] |- | 22 || 2025-03-28 || 2025-04-01 || 2025-04-05 || [[Trump ordered changes in public data]] |- | 21 || 2025-03-06 || 2025-03-11 || 2025-03-22 || [[Vulture capitalists destroying newspapers]] |- | 20 || 2025-02-25 || 2025-02-25 || 2025-03-08 || [[Local newspapers limit malfeasance]] |- | 19 || 2025-02-06 || 2025-02-11 || 2025-02-22 || [[Palast says Trump lost, vote suppression won the 2024 elections]] |- | 18 || 2025-01-25 || 2025-02-04 || 2025-02-12 || [[Defend free speech hybrid town hall]] |- | 17 || 2025-01-13 || 2025-01-14 || 2025-01-25 || [[Media in the Syrian conflict]] |- | 16 || 2024-12-20 || 2024-12-31 || 2024-01-04 || [[HR 9495, the nonprofit-killer bill, per Michael Novick]] |- | 15 || 2024-12-13 || 2024-12-24 || 2024-12-21 || [[Information is a public good per communications prof Pickard]] |- | 14 || 2024-12-02 || 2024-12-10 || 2024-12-07 || [[Media literacy for the Arab World per Ahmed Al-Rawi]] |- | 13 || 2024-11-21 || 2024-11-26 || 2024-11-23 || [[Thom Hartmann on The Hidden History of the American Dream]] |- | 12 || 2024-10-25 || 2024-11-05 || 2024-11-09 || [[Legal concerns of Wikimedia Europe]] |- | 11 || 2024-10-26 || 2024-10-19 || 2024-10-27 || [[Project 2025 per Professor Brooks]] |- | 10 || 2024-10-01 || 2024-10-01 || 2024-10-12 || [[Jacob Ware on far-right terrorism in the US]] |- | 9 || 2024-09-13 || 2024-09-17 || 2024-09-29 || [[Dis- and misinformation and their threats to democracy]] |- | 8 || 2024-09-11 || 2024-11-12 || 2024-09-14 || [[22nd Century Initiative]] |- | 7 || 2024-08-22|| 2024-08-27 || 2024-08-31 || [[Global Project Against Hate & Extremism (GPAHE)]] |- | 6 || 2024-08-19 || 2024-08-20 || 2024-08-24 || [[Facebook whistleblower Frances Haugen says]] |- | 5 || 2024-08-13 || 2024-08-13 || 2024-08-17 || [[Legal concerns of Free Press including Section 230]] |- | 4 || 2024-08-02 || 2024-08-06 || 2024-08-10 || [[How psychological and interpersonal processes are influenced by human-computer interactions]] |- | 3 || 2024-07-30 || 2024-07-30 || 2024-08-03 || [[Dean Baker on Internet companies threatening democracy internationally and how to fix that]] |- | 2 || 2021-04-29 || 2021-04-29 || 2021-05-16 || [[Media reform per Freepress.net]] |- | 1 || 2021-02-23 || 2021-02-23 || 2021-03-17 ||[[Unrigging the media and the economy]] |} == Notes == {{reflist}} == Bibliography == * <!-- H. R. McMaster (2020) Battlegrounds: The Fight to Defend the Free World-->{{cite Q|Q104774898}} * <!--Maria Ressa (2022) How to Stand Up To a Dictator-->{{cite Q|Q117559286}} [[Category:Interdisciplinary studies]] [[Category:Political science]] [[Category:Economics]] [[Category:Freedom and abundance]] [[Category:Videoconferences on media and democracy]] kvk2uvgifpdgjcngkdc9knj3q80kpou 2806701 2806700 2026-04-26T19:41:01Z DavidMCEddy 218607 /* Table of episodes */ typo 2806701 wikitext text/x-wiki A number of seemingly credible sources are describing an increase in political polarization worldwide. [[w:Maria Ressa|Maria Ressa]] describes how [[w:Rodrigo Duterte|Rodrigo Duterte]], former President of the [[w:Philippines|Philippines]] "started ... with five hundred volunteers<ref>Ressa (2022, pp. 147-8).</ref> (1) creating “sock puppets,” or fake accounts that attack or praise; (2) “mass reporting,” or organizing to negatively impact a targeted account; and (3) “astroturfing,” or fake posts or lies designed to look like grassroots support or interest.<ref>Ressa (2022, pp. 152-3).</ref> These actions tricked the algorithms of social media companies like Facebook and Twitter into amplifying fraudulent messages including incitements to violence and criminal prosecutions based on trumped up charges. The results easily overwhelmed honest media. [[w:Leila de Lima|Leila de Lima]], a Senator and former Secretary of Justice of the Philippines, spent years in pretrial detention before the charges were dropped for lack of evidence.<ref>Ressa (2022, p. 158ff) and Wikipedia, "[[w:Leila de Lima|Leila de Lima]]", accessed 2024-07-22.</ref> Ressa's news organization, [[w:Rappler|Rappler]].com, was ordered to close. Ressa herself was convicted on questionable charges. Both continued operating while the legal procedures against them were appealed.<ref>Ressa (2022, pp. 152-3) and Wikipedia, "[[w:Maria Ressa|Maria Ressa]]", accessed 2024-07-22.</ref> Ressa says similar procedures are making major contributions to the rise of fascism and far-right nationalist populists in the US, Europe, Turkey, India, Russia, and elsewhere.<ref>Ressa (2022, pp. 152-3).</ref> [[w:H. R. McMaster|H. R. McMaster]], former President Trump's second National Security advisor, said that "The internet and social media thus provided [Russia] with a low-cost, easy way to divide and weaken America from within."<ref>McMaster (2020, pp. 47-48).</ref> The [[w:2021 Facebook leak|2021 Facebook leak]] documented how executives of [[w:Facebook|Facebook]] and [[w:Meta Platforms|Meta]] knowingly prioritized profits over action to limit incitements to violence, even facilitating the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise would have reduced their profits. This "Category:Media reform to improve democracy" include videos of experts and activists working this issue along with 29:00 mm:ss audio files submitted to a ''Media & Democracy'' series syndicated on the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]<ref><!--Media & Democracy on Audioport-->{{cite Q|Q127839818}}</ref> plus text and space for moderated discussions. Some of this work is cited in the book on ''[[Media Literacy and You]]'', which is being written -- [[w:Crowdsourcing|crowdsourced]] -- to help humans better understand how they can counter the trend toward increasing political polarization and violence by talking politics, calmly, with respect and humility, with others with whom they may vehemently disagree, because the alternative is killing humans over misunderstanding. The goal is ''not'' to convince anyone that they are wrong. Rather it is to build relationships where humans can agree to disagree agreeably and collaborate to improve issues of common concern. == Table of episodes == {| class="wikitable sortable" |+ Episodes of "Media & Democracy" for the [[w:List of Pacifica Radio stations and affiliates|Pacifica Radio Network]] |- ! !! colspan=3 | Date !! |- ! no. || recorded !! broadcasted on [[w:KKFI|KKFI]] !! released to Pacifica !! Episode |- | 50 || 2026-04-09 || 2026-04-28 || 2026-05-02 || [[How US media threaten the health of all]] |- | 49 || 2026-04-06 || 2026-04-14 || 2026-04-18 || [[News suppressed for those who control money for the media]] |- | 48 || 2026-03-27 || 2026-03-31 || 2026-04-04 || [[Media and war]] |- | 47 || 2026-03-12 || 2026-03-17 || 2026-03-21 || [[Media literacy to dispel myths and improve public policy]] |- | 46 || 2026-02-26 || 2026-03-03 || 2026-03-07 || [[Concerns about media, especially in Germany]] |- | 45 || 2026-02-12 || 2026-02-17 || 2026-02-21 || [[Underserved serve themselves with low-power FM]] |- | 44 || 2026-01-30 || 2026-02-03 || 2026-02-07 || [[Conservative media are different]] |- | 43 || 2026-01-15 || 2026-01-20 || 2026-01-24 || [[Medill says you can help yourself by helping improve local media]] |- | 42 || 2026-01-03 || 2026-01-06 || 2026-01-10 || [[Lisa Loving on media literacy and how you can report for your community]] |- | 41 || 2015-12-11 || 2025-12-23 || 2025-12-27 || [[John Maxwell Hamilton on American propaganda]] |- | 40 || 2025-12-05 || 2025-12-09 || 2025-12-13 || [[You can better protect yourself from Big Tech]] |- | 39 || 2025-11-20 || 2025-11-25 || 2025-11-29 || [[Differences between media outlets including coverage of Gaza]] |- | 38 || 2025-11-06 || 2025-11-11 || 2025-11-15 || [[Media & Democracy lessons for the future]] |- | 37 || 2025-10-23 || 2025-20-28 || 2025-11-01 || [[Media reform initiatives in West Africa]] |- | 36 || 2025-10-03 || 2025-10-14 || 2025-10-18 || [[Seth Radwell says that the two Enlightenments tell us how to heal US political polarization]] |- | 35 || 2025-09-25 || 2025-09-30 || 2025-10-04 || [[Media Reform Coalition challenges anti-democratic media bias in the UK]] |- | 34 || 2025-09-12 || 2025-09-16 || 2025-09-20 || [[Fighting back against the campaign of censorship and control]] |- | 33 || 2025-08-28 || 2025-09-02 || 2025-08-06 || [[The role of the media in conflict]] |- | 32 || 2025-07-31 || 2025-08-19 || 2025-08-21 || [[Evidence-informed public policy]] |- | 31 || 2025-08-01 || 2025-08-05 || 2025-08-09 || [[What the Left can learn from Fox]] |- | 30 || 2025-07-17 || 2025-07-22 || 2025-07-26 || [[Democratic delusions: Fix the media to fix democracy]] |- | 29 || 2025-07-03 || 2025-07-08 || 2025-07-12 || [[News from Germany 1900-1945 and implications for today]] |- | 28 || 2025-06-12 || 2025-06-24 || 2025-06-28 || [[How news impacts democracy per USD Communications Professor Nik Usher]] |- | 27 || 2025-06-08 || 2025-06-10 || 2025-06-14 || [[Media concentration per Columbia History Professor Richard John]] |- | 26 || 2025-05-21 || 2025-05-27 || 2025-05-31 || [[Dean Starkman and the watchdog that didn't bark]] |- | 25 || 2025-05-08 || 2025-05-13 || 2025-05-17 || [[Freedom of the Press Foundation says...]] |- | 24 || 2025-04-24 || 2025-04-29 || 2025-05-03 || [[Canadian journalist Marc Edge on media reform to improve democracy]] |- | 23 || 2025-04-10 || 2025-04-15 || 2025-04-19 || [[The value of indigenous and community radio]] |- | 22 || 2025-03-28 || 2025-04-01 || 2025-04-05 || [[Trump ordered changes in public data]] |- | 21 || 2025-03-06 || 2025-03-11 || 2025-03-22 || [[Vulture capitalists destroying newspapers]] |- | 20 || 2025-02-25 || 2025-02-25 || 2025-03-08 || [[Local newspapers limit malfeasance]] |- | 19 || 2025-02-06 || 2025-02-11 || 2025-02-22 || [[Palast says Trump lost, vote suppression won the 2024 elections]] |- | 18 || 2025-01-25 || 2025-02-04 || 2025-02-12 || [[Defend free speech hybrid town hall]] |- | 17 || 2025-01-13 || 2025-01-14 || 2025-01-25 || [[Media in the Syrian conflict]] |- | 16 || 2024-12-20 || 2024-12-31 || 2025-01-04 || [[HR 9495, the nonprofit-killer bill, per Michael Novick]] |- | 15 || 2024-12-13 || 2024-12-24 || 2024-12-21 || [[Information is a public good per communications prof Pickard]] |- | 14 || 2024-12-02 || 2024-12-10 || 2024-12-07 || [[Media literacy for the Arab World per Ahmed Al-Rawi]] |- | 13 || 2024-11-21 || 2024-11-26 || 2024-11-23 || [[Thom Hartmann on The Hidden History of the American Dream]] |- | 12 || 2024-10-25 || 2024-11-05 || 2024-11-09 || [[Legal concerns of Wikimedia Europe]] |- | 11 || 2024-10-26 || 2024-10-19 || 2024-10-27 || [[Project 2025 per Professor Brooks]] |- | 10 || 2024-10-01 || 2024-10-01 || 2024-10-12 || [[Jacob Ware on far-right terrorism in the US]] |- | 9 || 2024-09-13 || 2024-09-17 || 2024-09-29 || [[Dis- and misinformation and their threats to democracy]] |- | 8 || 2024-09-11 || 2024-11-12 || 2024-09-14 || [[22nd Century Initiative]] |- | 7 || 2024-08-22|| 2024-08-27 || 2024-08-31 || [[Global Project Against Hate & Extremism (GPAHE)]] |- | 6 || 2024-08-19 || 2024-08-20 || 2024-08-24 || [[Facebook whistleblower Frances Haugen says]] |- | 5 || 2024-08-13 || 2024-08-13 || 2024-08-17 || [[Legal concerns of Free Press including Section 230]] |- | 4 || 2024-08-02 || 2024-08-06 || 2024-08-10 || [[How psychological and interpersonal processes are influenced by human-computer interactions]] |- | 3 || 2024-07-30 || 2024-07-30 || 2024-08-03 || [[Dean Baker on Internet companies threatening democracy internationally and how to fix that]] |- | 2 || 2021-04-29 || 2021-04-29 || 2021-05-16 || [[Media reform per Freepress.net]] |- | 1 || 2021-02-23 || 2021-02-23 || 2021-03-17 ||[[Unrigging the media and the economy]] |} == Notes == {{reflist}} == Bibliography == * <!-- H. R. McMaster (2020) Battlegrounds: The Fight to Defend the Free World-->{{cite Q|Q104774898}} * <!--Maria Ressa (2022) How to Stand Up To a Dictator-->{{cite Q|Q117559286}} [[Category:Interdisciplinary studies]] [[Category:Political science]] [[Category:Economics]] [[Category:Freedom and abundance]] [[Category:Videoconferences on media and democracy]] 052rfls6xj2ehmfd9nha93xhzf9h3xo 0 285380 2806775 2806550 2026-04-27T10:12:13Z Young1lim 21186 /* Applications */ 2806775 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260427.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> egcl0eg2w5joqgujugn9f0b39497upk 0 292498 2806767 2788140 2026-04-27T04:46:21Z RadiX 155307 upd 2806767 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global_Audiology/Americas/Header}} {{CountryHeader|File:BRA orthographic.svg|https://en.wikipedia.org/wiki/Brazil}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Brazil Brazil], officially known as the Federative Republic of Brazil, is the largest country in Latin America. Covering roughly half of South America's land area, it borders all other countries and territories on the continent except Ecuador and Chile. The official language of Brazil is Portuguese, but one hundred and eighty Amerindian languages are spoken in remote areas and a significant number of other languages are spoken by immigrants and their descendants. {{HTitle|History of Audiology}} Due to the concern of professionals in medicine and education regarding the prevention and treatment of communication disorders among schoolchildren, Brazil identified the need for a profession to provide care for conditions affecting speech, language, and hearing as early as the 1930s. The term ''Fonoaudiologia'', in Portuguese, was coined to include speech, language, and hearing conditions in its scope. In the 1960s, speech pathology and audiology courses in Brazil began with the first course at the technical level at the University of São Paulo (1961), by the Otorhinolaryngology Department of the Clinical Hospital of the Faculty of Medicine, and by the Pontifical Catholic University of São Paulo (1962) from its Institute of Psychology. The first bachelor’s degree course in speech therapy started in 1971 at the Federal University of Santa Maria, in Rio Grande do Sul. <ref>{{Cite journal|last=Berberian|first=Ana Paula|date=2001-06|title=Speech Pathology and Audiology: A historical analysis|journal=Dist Comun|volume=12|issue=2|}}</ref> Sanctioned on December 9, 1981, by President João Figueiredo, Law number 6965, regulated the profession of speech pathology and audiology (in Portuguese ''Fonoaudiólogo''). In addition to identifying the competencies of this profession, Federal and Regional Councils were created with the purpose of supervising the professional practice. Since 1981, the 4-year bachelor’s degree became the entry-level degree for clinical practice. The first minimum coursework load (3,700 hours), which identified required disciplines, was regulated by Resolution No. 54/76 of the Federal Council of Education. Coursework includes disciplines related to the biological sciences and the health area, such as anatomy, physiology, genetics, and pathologies. As it pertains to social and human sciences, the student takes classes in psychology, pedagogic methods, and ethics. However, most of the training is devoted to specific content of the profession, such as the auditory system, oral and written language, or speech. Supervised clinical experience and the completion of a monograph are required. The activities of the Federal Speech and Hearing Therapy Council [https://fonoaudiologia.org.br/cffa-abre-edital-para-contratacao-de-empresa-especializada-em-assessoria-parlamentar/ (CFFa)] began in 1983. On September 15, 1984, by Resolution CFFA No. 010/84, the first Code of Ethics of the profession was approved, which included the rights, duties, and responsibilities inherent to the various relationships established for professional activity. The CFFA exercises a normative function by publishing resolutions that aim to establish interpretations that facilitate the execution of the provisions of the law, according to professional reality and technical-scientific advancements. All resolutions come into force after publication in the Official Gazette of the Union. The Regional Councils have the priority role of enforcing the provisions of the law, the Professional Code of Ethics, and the norms of the Federal Council and directing and supervising professional practice in their area of jurisdiction. For this purpose, the Regional Council is responsible for the issuance of professional records, the establishment of ethical-disciplinary and/or administrative processes, the adjudication of infractions, and applying the penalties provided for in the law. {{HTitle|Incidence and Prevalence of Hearing Loss}} According to the 2010 census ([https://censo2010.ibge.gov.br/ Demographic Census, 2010]), there are 9.8 million individuals with hearing impairment in the country; however, the accuracy of this information must be considered in terms of time and the methodology used, which is a self-report questionnaire. Although there is a shortage of specific statistical data on hearing impairment in Brazil, according to the World Health Organization (WHO), 1.5% of the Brazilian population would have some degree of hearing impairment (WHO, 1999). An analysis of different epidemiological studies in audiology in Brazil, published in 2011, concluded that there is greater concern regarding hearing disorders related to their occupation. <ref>{{Cite journal|last=Arakawa|first=Aline Megumi|last2=Sitta|first2=Érica Ibelli|last3=Caldana|first3=Magali de Lourdes|last4=Sales-Peres|first4=Sílvia Helena de Carvalho|date=2010-08-13|title=Análise de diferentes estudos epidemiológicos em Audiologia realizados no Brasil|url=http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1516-18462011000100018&lng=pt&tlng=pt|journal=Revista CEFAC|volume=13|issue=1|pages=152–158|doi=10.1590/S1516-18462010005000089|issn=1982-0216}}</ref> Considering that Brazil is a country the size of a continent, there are regional particularities that may influence the population’s general health profile, including hearing health, prevalence of tropical diseases in Northern areas, cultural characteristics which determine different habits of overall health care, and the logistical difficulties imposed by the geographic and economic characteristics of each region. This reality is reflected in the only two population-based studies using the World Health Organization Ear and Hearing Disorders Survey Protocol in opposite regions of the country. In 2003, the urban area of Canoas, Southern Brazil had the prevalence of disabling hearing impairment of 6.8%, prevalence of moderate hearing impairment of 5.4%, severe hearing impairment of 1.2%, profound hearing impairment of 0.2%, and slight hearing impairment of 19.3%. <ref>{{Cite journal|last=Béria|first=Jorge Umberto|last2=Raymann|first2=Beatriz Carmen Warth|last3=Gigante|first3=Luciana Petrucci|last4=Figueiredo|first4=Andréia Cristina Leal|last5=Jotz|first5=Geraldo|last6=Roithman|first6=Renato|last7=Selaimen da Costa|first7=Sady|last8=Garcez|first8=Vera|last9=Scherer|first9=Caroline|date=2007-06|title=Hearing impairment and socioeconomic factors: a population-based survey of an urban locality in southern Brazil|url=https://pubmed.ncbi.nlm.nih.gov/17761050|journal=Revista Panamericana De Salud Publica = Pan American Journal of Public Health|volume=21|issue=6|pages=381–387|doi=10.1590/s1020-49892007000500006|issn=1020-4989|pmid=17761050}}</ref> On the other hand, in the urban area of Montenegro city, RO, of 2005 to 2007, the results showed that 3.8% of population were classified in the disabling hearing impairment category. The prevalence of moderate hearing impairment was 3.4%, severe hearing impairment was 0.4%, and profound hearing impairment was not found.<ref>{{Cite journal|last=Bevilacqua|first=Maria Cecilia|last2=Banhara|first2=Marcos Roberto|last3=Oliveira|first3=Ariádnes Nóbrega de|last4=Moret|first4=Adriane Lima Mortari|last5=Alvarenga|first5=Kátia de Freitas|last6=Caldana|first6=Magali de Lourdes|last7=Camargo|first7=Luís Marcelo Aranha|last8=Costa|first8=Orozimbo Alves|last9=Bastos|first9=José Roberto de Magalhães|date=2013-04|title=Survey of hearing disorders in an urban population in Rondonia, Northern Brazil|url=https://pubmed.ncbi.nlm.nih.gov/24037358|journal=Revista De Saude Publica|volume=47|issue=2|pages=309–315|doi=10.1590/S0034-8910.2013047004059|issn=1518-8787|pmid=24037358}}</ref> Different prevalence of congenital sensorineural hearing loss ranging from 2.3:1000 live births<ref>{{Cite journal|last=Chapchap|first=M. J.|last2=Segre|first2=C. M.|date=2001|title=Universal newborn hearing screening and transient evoked otoacoustic emission: new concepts in Brazil|url=https://pubmed.ncbi.nlm.nih.gov/11409775|journal=Scandinavian Audiology. Supplementum|issue=53|pages=33–36|doi=10.1080/010503901750166600|issn=0107-8593|pmid=11409775}}</ref> to 0.96:1000 have also been reported.<ref>{{Cite journal|last=Bevilacqua|first=Maria Cecilia|last2=Alvarenga|first2=Kátia de Freitas|last3=Costa|first3=Orozimbo Alves|last4=Moret|first4=Adriane Lima Mortari|date=2010-05|title=The universal newborn hearing screening in Brazil: from identification to intervention|url=https://pubmed.ncbi.nlm.nih.gov/20303604|journal=International Journal of Pediatric Otorhinolaryngology|volume=74|issue=5|pages=510–515|doi=10.1016/j.ijporl.2010.02.009|issn=1872-8464|pmid=20303604}}</ref> The ''Comitê Multiprofissional em Saude Auditiva'' (COMUSA) advocates that the diagnosis of hearing impairment should be made up to three months of age and intervention initiated up to six months, an internationally similar recommendation.<ref>{{Cite journal|last=Lewis|first=Doris Ruthy|last2=Marone|first2=Silvio Antonio Monteiro|last3=Mendes|first3=Beatriz C. A.|last4=Cruz|first4=Oswaldo Laercio Mendonça|last5=Nóbrega|first5=Manoel de|date=2010|title=Multiprofessional committee on auditory health: COMUSA|url=https://pubmed.ncbi.nlm.nih.gov/20339700|journal=Brazilian Journal of Otorhinolaryngology|volume=76|issue=1|pages=121–128|doi=10.1590/S1808-86942010000100020|issn=1808-8686|pmc=9446045|pmid=20339700}}</ref> Therefore, it is important for early detection of this condition through newborn hearing screening, a procedure capable of detecting and intervening early in hearing deficits that may interfere with the individual’s life. Such programs have been developed since the 80’s with populations of high and low risk factors, using behavioral and electrophysiological procedures. Nowadays, all newborns are expected to go through neonatal hearing screening instead of only those with a risk indicator for hearing loss. The [https://www.conass.org.br/ci-n377-publicada-portaria-sas-n1328-que-que-ficam-aprovadas-as-diretrizes-de-atencao-a-triagem-auditiva-neonatal/ Ordinance GM / MS no. 1,328] of December 3, 2012 pertaining to the [https://www.gov.br/mdh/pt-br/assuntos/noticias/2023/novembro/PlanoNacionaldosDireitosdaPessoacomDeficinciaNovoViverSemLimite.pdf care network for people with disabilities] addresses the Guidelines for Attention to Newborn Hearing Screening under the Brazilian Unified Health System ([https://www.gov.br/saude/pt-br/sus SUS – ''Sistema Único de Saúde'']). [https://www.gov.br/saude/pt-br/assuntos/saude-de-a-a-z/s/saude-da-pessoa-com-deficiencia/publicacoes/diretrizes-de-atencao-da-triagem-auditiva-neonatal.pdf Universal Neonatal Hearing Screening] is considered the main strategy to identify neonates and infants, in order to achieve early diagnosis and intervention which are crucial for the acquisition of oral language in these children. There is no official data on the total number of maternity hospitals that have the neonatal hearing screening program in Brazil. [https://assets.website-files.com/5d7f96ea4cc8598434877fed/5d7f96ea4cc8590706878100_Saude_Brazil_2018-compressed.pdf The Ministry of Health Management Report] (2018), established the goal to equip 308 non-profit maternity hospitals and 259 public administration maternity hospitals by 2019. However, despite government efforts, the goal of universal neonatal hearing screening has not yet been achieved. In 2015 only 37.2% of newborns were tested, with no significant increase from previous years.<ref>{{Cite journal|last=Paschoal|first=Monique Ramos|last2=Cavalcanti|first2=Hannalice Gottschalck|last3=Ferreira|first3=Maria Ângela Fernandes|date=2017-11|title=[Spatial and temporal analysis of the coverage for neonatal hearing screening in Brazil (2008-2015)]|url=https://pubmed.ncbi.nlm.nih.gov/29211167|journal=Ciencia & Saude Coletiva|volume=22|issue=11|pages=3615–3624|doi=10.1590/1413-812320172211.21452016|issn=1678-4561|pmid=29211167}}</ref> Ineffective referral and counter-referral systems, along with a limited adherence of families to the different stages of the child hearing health program, hinder the prevention, identification, and treatment of hearing loss at the ideal age. The early identification of hearing loss should not be exclusively linked to neonatal hearing screening programs. Thus, the World Health Organization and the Ministry of Health (2012) have recommended the collaboration with primary health care, which will enable the identification of acquired or progressive hearing losses, as early as possible, minimizing their impact on child development. In this proposal, the Community Health Agent has an extremely important role due to the scope of the Family Health Strategy and the close relationship between the  professional and the families.<ref>{{Cite journal|last=Oliveira|first=Maria Taiany Duarte de|last2=Alvarenga|first2=Kátia de Freitas|last3=Amorim|first3=Alice Andrade Lopes|last4=Jacob|first4=Lilian Cassia Bornia|last5=Araújo|first5=Eliene Silva|date=2023|title=Analysis of a hearing loss identification and intervention program in the first years of life in primary care|url=http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1516-18462023000100507&tlng=en|journal=Revista CEFAC|volume=25|issue=1|doi=10.1590/1982-0216/20232518522|issn=1982-0216}}</ref> According to the primary health care indicator panels, using April 2020 as a reference, Brazil had 44,716 Family Health Strategy teams which reflects an estimate of 65.4% coverage or 137,360,577 people. This reflects an increase of 76.5% (160,780,129 people) when considering primary care as a whole (Aps Indicator Panels, n.d.).{{citation needed}} The continuous growth of coverage by the Family Health Strategy and primary care throughout the Brazilian territory is notorious; however, there are differences in coverage when considering the regions, states, and municipalities of the country. There is a large number of people without access to the proposed monitoring. In Brazil, studies present a variety of prevalence of hearing loss between 24% to 45% in the elderly population showing a higher prevalence in men.<ref>{{Cite journal|last=Mattos|first=Leila Couto|last2=Veras|first2=Renato Peixoto|date=2007|title=The prevalence of hearing loss in an elderly population in Rio de Janeiro: a cross-sectional study|url=https://pubmed.ncbi.nlm.nih.gov/18094807|journal=Brazilian Journal of Otorhinolaryngology|volume=73|issue=5|pages=654–659|doi=10.1016/s1808-8694(15)30126-9|issn=1808-8694|pmc=9445649|pmid=18094807}}</ref> Brazil is undergoing a demographic shift with an increase in the proportion of older adults in the population composition which is likely to impact the overall rates of hearing conditions and demand for audiological services (''Population Projections | IBGE'', n.d.). In 2043, a quarter of the population is expected to be over 60 years old while the proportion of young people up to 14 years old will be only 16.3%. The aging rate is expected to increase from 43.19% in 2018 to 173.47% in 2060. Because of the growing aging of the population, the journal specialized in careers reports that there will be a growth of 34% in the labor market for hearing care professionals. For professionals working in this sector, job opportunities continue to grow due to factors such as advances in public health policies, coverage of medical covenants, aging of the population, and modern technological resources used in the various areas of health care. {{HTitle|Information About Audiology}} === Educational Institutions === Since its recognition by the Federal Government (Law n°6965), the undergraduate course in speech pathology and audiology “aims to train professionals qualified to work in the two areas." The availability of graduate training not only raised the standard of professional care but also expanded professional opportunities and the scope of practice for audiologists in the country. In 2020, there were 87 undergraduate courses in speech pathology and audiology recognized by the Ministry of Education. Of those, 24 (27.6%) are offered by public universities, while 63 (72.4%) are offered by private ones. In addition, at the graduate level, different courses are offered in the four areas of specialization defined by the [https://fonoaudiologia.org.br/fonoaudiologos/ensino-superior/%20Conselho%20Federal%20de%20Fonoaudiologia%20(CFFa) Speech-Language Pathology and Audiology Council] (CFFA, audiology, language, oral motor disorders, and voice). Therefore, speech pathologists and audiologists are health professionals who specialize in the identification, diagnosis, treatment, prevention, and monitoring of communication disorders and orofacial functions. The CFFAhas the current universities that offer speech-language pathology and audiology courses. === Audiology Practice: Public & Private === The speech pathologists and audiologists who work in the audiology area (audiologists) perform diagnostic evaluations of auditory and vestibular disorders, select and fit hearing aids, map cochlear implants, and provide auditory rehabilitation. Currently, audiologists in Brazil work at public and private establishments. The public institutions include community clinics, elementary schools, colleges, hospitals, industries, and universities. ==== Unified Health System ==== Universal and equal access to healthcare is one of the guiding principles of Brazil’s Unified Health System (''Sistema Único de Saúde'': SUS). The SUS is organized into three levels of care, and audiological services are available at each level. Each successive level involves services of increasing complexity. All levels work together in an organized network of assistance, referral, and counter-referral. Priorities are defined in health conferences, and supervising councils make decisions. In 1993, the Unified Health System began to reimburse the following procedures related to the diagnosis of hearing loss: adaptation of individual hearing aids, speech therapy, and necessary follow-up after adaptation. In 2000, these procedures were regulated by an ordinance, and later in 2004, the procedures were organized by the National Policy of Attention to Hearing Health by the Ministry of Health into basic, medium, and high complexity care to better organize the patient flow and its distribution in the network. This ordinance ensures citizens' access to the Hearing Health Program, which aims to develop actions to promote, prevent, intervene, and develop hearing health care. ==== Basic Attention (Primary Health Care) ==== This level includes low-cost health service technology. It involves an interdisciplinary team working in conjunction with the patient and the family. In the area of hearing loss, the Unified Health System's services at the basic level include the provision of information and community guidance to assist in the early identification of hearing problems, public health programs to prevent the main causes of avoidable hearing impairment, and identification of community resources for the person with hearing impairment. Currently, the basic attention system is examining a proposal to have providers ask the basic question, "Does your child hear well?" in every interaction with parents. Orienting families to the hearing health of their children emphasizes the importance of early detection of hearing loss and encourages parents to monitor speech, language, and hearing milestones and report any concerns.<ref>{{Cite journal|last=Bevilacqua|first=Maria Cecilia|last2=Novaes|first2=Beatriz Caiuby|last3=Morata|first3=Thais C.|date=2008-02|title=Audiology in Brazil|url=https://pubmed.ncbi.nlm.nih.gov/18236235|journal=International Journal of Audiology|volume=47|issue=2|pages=45–50|doi=10.1080/14992020701770843|issn=1708-8186|pmid=18236235}}</ref> ==== Secondary Level ==== This level involves the operation of public clinics where professionals offer diagnostic and rehabilitative services, provide technical support to basic attention level teams, and identify and refer cases that require higher level services. In regards to hearing health, secondary level programs include ENT and audiological evaluations, such as hearing screening in newborns, preschoolers, and school-aged children; audiometric monitoring of noise-exposed individuals; aural rehabilitation; speech-language pathology evaluations and therapy; and hearing-aid selection and fitting. Related services include psychological assessment and therapy, social work assistance, family and school orientations, and home and/or institutional visits. Secondary level service providers inform basic attention teams of the main causes of hearing loss, methods for prevention, and methods for early identification of hearing problems (Bevilacqua et al., 2008). ==== High Complexity (Tertiary Level) ==== This service level provides advanced diagnostics and treatment, as well as basic care, to difficult-to-treat populations. Additionally, this level is responsible for the qualification of basic attention level personnel and oversight of services provided at the secondary level. In the realm of hearing health, tertiary level teams provide sophisticated testing services, including otoacoustic emissions evaluation (distortion-product and transient-evoked) and auditory evoked potential testing (including auditory brainstem responses and middle and long latency potentials). They also provide testing and hearing-aid services to children under age three and patients with multiple disorders. In 2011, this systematic action was changed by another government policy called ''[https://www.gov.br/mdh/pt-br/assuntos/noticias/2023/novembro/PlanoNacionaldosDireitosdaPessoacomDeficinciaNovoViverSemLimite.pdf Plano Viver sem Limite]'' in an attempt to improve access to care and promote citizenship. In this plan, new specialized rehabilitation centers were created that must serve a minimum of 2 of the following areas: auditory, motor, visual, or cognitive. === Services Offered by Otorhinolaryngologists & Otolaryngologists === Otorhinolaryngology (ENT) is a clinical-surgical medical specialty that treats problems related to the ear, nose, sinuses, and throat through medications, surgical procedures, or rehabilitation. An ENT, or otolaryngologist, is a physician who has specialized in caring for the ears, nose, and throat. The main difference between ENTs and audiologists is that an ENT is a medical ear doctor, while an audiologist is a professional hearing doctor without a medical degree. ENTs are trained to perform surgery on the ears, nose, and throat. They can also prescribe medication. They usually handle conductive hearing loss issues in adults and children with devices such as cochlear implants. Audiologists work on sensorineural hearing loss, which affects the inner ear. Other conditions ENTs treat are hearing loss due to ear trauma, infection, and benign tumors. Once ENTs perform surgery and prescribe necessary medications, they often refer patients to an audiologist for continued care, especially if rehabilitation is part of the aftercare. === Audiological Services === After any determination of hearing loss or vestibular abnormality is made, the professional will provide recommendations to the patient (for example, hearing aids, cochlear implants, or an appropriate medical referral). They are trained to diagnose, manage, and/or treat hearing, tinnitus, or balance problems. They promote, manage, and adapt hearing aids, evaluate the potential application, and map cochlear implants. They advise families when there is a new diagnosis of hearing loss in infants and provide coping and compensation skills for adults who become deaf. Audiologists also help develop and implement personal and industrial hearing safety programs, newborn hearing screening programs, hearing screening programs for school-aged children, and provide special fittings for ear plugs and other hearing protection devices to help prevent hearing loss. Audiologists are trained to evaluate peripheral vestibular disorders arising from internal ear pathologies. They also offer treatment for certain vestibular and balance disorders. In addition, many work as scientists developing and innovating new techniques and approaches to treat individuals with vestibular and audiologic conditions. === Professionals === The salaries of speech pathologists and audiologists in Brazil vary by the setting of the workplace. Taking into account purchasing power parity (PPP) and the cost of living in different countries, the average wage across countries is US $1,480, and in Brazil, US $778 (Gonçalves et al. 2014) for those who work in the public sector. Higher incomes are possible from jobs in the private sector. === Professional and Regulatory Bodies === The activities and movements for professional and course recognition started in the 1970s. Subsequently, bachelor’s courses were established. The first was at the University of São Paulo, authorized in 1977. In 1981, the federal government formally recognized the profession through Legislative Acts [https://www.planalto.gov.br/ccivil_03/leis/l6965.htm 6965/81] and [https://www2.camara.leg.br/legin/fed/decret/1980-1987/decreto-87218-31-maio-1982-436966-publicacaooriginal-1-pe.html|87218/82 Decree 87.218]. In addition to determining the competence of the profession, the Federal and Regional Councils of Speech-Language Pathology and Audiology were created with the main purpose of supervising professional practice. The activities of the Federal Council started in 1983 and the following year the first Code of Ethics of the profession was published. This document presents the rights, duties and responsibilities of the profession (Resolution CFFa No. 010/84). The progress and the expansion allowed awareness of the class. The Federal Council has a normative purpose, issuing resolutions designed to regulate the profession according to the Federal Law (Constitution) and technical-scientific progress. The Regional Councils are responsible for applying and inspecting the Ethical Code and the Federal Council rules. They also guide and supervise professional practice in their jurisdiction. Examples would be the issuance of professional records, the establishment of ethical-disciplinary and/or administrative processes, and the judgment of infractions. As mentioned before, there are nine councils in Brazil that cover the different states of the country. According to [https://www.planalto.gov.br/ccivil_03/leis/l6965.htm Law 6965/81], an annual payment to the Regional Council of the respective jurisdiction constitutes a condition of legitimacy for the exercise of the profession. It is important to mention that in addition to the Councils in Brazil, the ''Classes Societies'' also exist. These classes are regulated by a statute which regulates the fundamental purposes of the institution and specifies its operation. In Brazil, the [https://fonoaudiologia.sbfa.org.br/ Brazilian Society of Language, Speech Therapy and Audiology] (SBFa), was created and lead by the Department of Hearing and Balance with support from the [https://audiologiabrasil.org.br/portal2018/ Brazilian Academy of Audiology] (ABA). === Scope of Practice and Licensing === In Brazil, speech pathology and audiology are interconnected; therefore, the profession developed as a unified field. A double license is granted after the four-year course, and the professional is allowed to practice in any area after complying with the federal regulatory standards. Audiology was established as one of the specialties of speech therapy among the other eleven different ones recognized by the Brazilian Federal Speech Language Pathology and Audiology Council. There are several options for degrees and coursework. Some courses offer a 500-hour course of study which provides advanced clinical training. In addition, some universities have master’s and doctoral degree courses designed to provide the necessary training for those aiming at a career in research and education. Audiology is perceived as the field related to promotion, prevention, diagnosis, and rehabilitation of the auditory and vestibular function, including research. The goal is guaranteeing the quality of life of the individual (CFFA, 2006). According to the Federal Council of Speech Therapy ([https://www.fonoaudiologia.org.br/resolucoes/resolucoes_html/CFFa_N_320_06.htm Resolution CFFa nº 320, of February 17, 2006]), the domain of the specialist in audiology includes knowledge in: # Strategies and programs to allow hearing health promotion. # Preventing and diagnosing auditory and vestibular dysfunctions. # Selection, adaptation, and monitoring of subjects with hearing aids, cochlear implants, or any other devices for hearing rehabilitation or hearing protection. # Therapy for (re)habilitation of hearing with devices and communicative strategies. {{HTitle|Research in Audiology}} The areas involving research are diverse in the field of audiology, starting from prevention and diagnosis to treatment (clinical and surgical intervention) of individuals with hearing loss in different age groups. In addition, several interdisciplinary research projects are linked to the public guidelines on hearing health. In Brazil, a governmental website entitled [http://dgp.cnpq.br/dgp/espelhogrupo/8082 Directory of Research Groups of the National Council for Scientific and Technological Development (CNPq)] is currently responsible for more than 40 registered research groups in audiology associated with public and private institutions in all regions of the country. The first research group in the area was created in 1990 and named the [http://dgp.cnpq.br/dgp/espelhogrupo/2746831243459381 Center for Audiological Research] (CPA). It worked in partnership between the Department of Speech Therapy at the [https://fob.usp.br/ Faculty of Dentistry of Bauru City (FOB)] and the Rehabilitation Hospital of Craniofacial Anomalies (HRAC). This collaboration promoted a significant increase in researchers from other centers and universities and allowed national and international collaborations, providing and developing multicenter research as well as inter-institutional academic agreements. Most of the research in audiology is supported by postgraduate programs (master and doctorate levels). At this time, the country has 12 postgraduate programs in the field of audiology disseminated in eight states from the country. {{HTitle|Audiology Charities}} There are several philanthropic organizations to assist families and individuals with hearing loss in Brazil; some of them are listed below: # [http://www.selosocial.com/apadas Association of Parents and Friends of the Hearing Impaired of Sorocaba (APADAS)] Created in 1988, their mission is to assist and promote research in the areas of prevention, diagnosis, and rehabilitation of patients with hearing loss associated with or without neurological and/or visual disorders. # [https://adap.org.br/ Association of the Hearing Impaired, Parents, and Friends (ADAP)] Established in 1998 in Bauru, Sao Paulo, by parents and cochlear implant recipients. The goal is to assist new patients and support the patients and families as they continue using the devices. # Association of Attention to the Hearing Impaired and Deaf – AADAS. Founded on May 24th, 1989 with the aim to serve children and adolescents by providing specialized care focused on quality of life and exercising citizenship with dignity. {{HTitle|Challenges, Opportunities and Notes}} As the field is constantly evolving, it is important to be prepared to continue studying. The advance of public health policies, the federal law that requires medical covenants to cover at least six sessions per year, the aging of the population, and the modern technological resources of medicine are some of the main factors that increase the demand for this profession. Opportunities for the profession can be found mainly in large urban centers. The best opportunities are in public settings and public health, where the professional manages and creates policies. {{HTitle|References}} {{reflist}} {{HTitle|Useful Links}} *[https://www.audiologiabrasil.org.br/ Brazilian Academy of Audiology] *[http://www.aborlccf.org.br/ Brazilian ENT Society] *[https://sbgg.org.br/ Brazilian Geriatric Gerontology Society] *[https://www.ibge.gov.br/ Brazilian Institute of Geography and Statistics] *[http://portalms.saude.gov.br/ Brazilian Ministry of Health] *[http://www.fonoaudiologia.org.br/ Federal Council for Speech Pathology and Audiologist] *[https://www.sbfa.org.br/portal2017/ Speech and Hearing Brazilian Society] {{Global Audiology Authors |name1=Lilian Felipe |role1=Author |website1=https://www.lamar.edu/fine-arts-communication/speech-and-hearing-sciences/faculty/lilian-felipe.html |name2=Lilian Jacob |role2=Author |researchgate2=https://www.researchgate.net/profile/Lilian-Corteletti-2 |name3=Katia Alvarenga de Freitas |role3=Author |researchgate3=https://www.researchgate.net/scientific-contributions/Katia-de-Freitas-Alvarenga-40070442 |name4=Eliene Silva Araujo |role4=Author |website4=https://docente.ufrn.br/201900363683/perfil |name5=Thais C. Morata |role5=Author |researchgate5=https://www.researchgate.net/profile/Thais-Morata |name6=Joyce Rodvie Sagun |role6=Contributor |linkedin6=http://linkedin.com/in/joyce-rodvie-sagun-4691bb182 }} {{:Global Audiology/footer}} [[Category:Brazil]] [[Category:Audiology]] m408qri6cmjl0bmryr1r5ppiszkefgi 0 292523 2806768 2804977 2026-04-27T04:56:56Z RadiX 155307 upd 2806768 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Asia/Header}} {{CountryHeader|File:India (orthographic projection).svg|https://en.wikipedia.org/wiki/India}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/India India], officially the Republic of India, is a country in South Asia. It is bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast, it shares land borders with Pakistan to the west; China, Nepal, and Bhutan to the north; and Bangladesh and Myanmar to the east. In the Indian Ocean, India is near Sri Lanka and the Maldives; its Andaman and Nicobar Islands share a maritime border with Myanmar, Thailand, and Indonesia. Hindi and English are the official languages, but it has a great linguistic diversity. The most commonly spoke languages are Hindi, Bengali, Telugu, Marathi, Tamil, and Urdu—are each spoken by more than 50 million people. {{HTitle|Hearing Loss Incidence and Prevalence}}In India, according to a recent study, 63 million people suffer from hearing loss (6.3).<ref>{{Cite journal|last=Varshney|first=Saurabh|date=2016|title=Deafness in India|url=http://www.indianjotol.org/text.asp?2016/22/2/73/182281|journal=Indian Journal of Otology|language=en|volume=22|issue=2|pages=73|doi=10.4103/0971-7749.182281|issn=0971-7749}}</ref> Of the total population of persons with disability of 2,68,14,994 in India, 50,72,914 persons are reported to have hearing loss (18.91%) across age groups. Among less than 6-year-old children, 4,76,075 children are reported to have some degree of hearing loss (23.20%) ([https://www.data.gov.in/catalog/primary-census-abstract-2011-india-and-states-0 Census Government of India], 2011). This number might be an understatement due to many unidentified hearing loss and unavailability of data (Paul, 2016; Sija et al., 2022).{{Citation needed}} The prevalence figures could be considered as a broad approximation, as these may include conductive, mild, and unilateral types of hearing loss in addition to permanent hearing losses. The most commonly used definition of hearing loss is as per the ([https://megscpwd.gov.in Rights of Persons with Disability], 2016), which states that; -“deaf” means persons having 70 dB hearing loss in speech frequencies in both ears & -“hard of hearing” means person having 60 dB to 70 dB hearing loss in speech frequencies in both ears {{HTitle|Information About Audiology}} === History === The profession of otolaryngology in ancient India can be dated back to between 1000 BC and 100 AD <ref>{{Cite journal|last=Pothula|first=V. B.|last2=Jones|first2=T. M.|last3=Lesser|first3=T. H. J.|date=2001-03|title=Otology in ancient India|url=https://www.cambridge.org/core/product/identifier/S0022215101000500/type/journal_article|journal=The Journal of Laryngology & Otology|language=en|volume=115|issue=3|pages=179–183|doi=10.1258/0022215011907091|issn=0022-2151}}</ref> ; however, otology as an independent profession gained more recognition after 1950.<ref>{{Cite journal|last=Bhargava|first=K. B.|last2=Bhargava|first2=S. K.|date=1996-04|title=Evolution of otology in India|url=https://link.springer.com/10.1007/BF03048052|journal=Indian Journal of Otolaryngology and Head and Neck Surgery|language=en|volume=48|issue=2|pages=93–95|doi=10.1007/BF03048052|issn=0019-5421}}</ref> The profession of audiology is relatively new and took root nearly half a century ago. Since then, both of these professions have undergone some major developments and offer a wide range of ear and hearing healthcare services. Influence of Western Medicine (American models) led to hearing healthcare services that include equipment-based hearing assessment and rehabilitation with the use of devices such as hearing aids. Indian traditional medicine, such as Ayurveda, Unani, and Siddha, has some treatment solutions for ear and hearing problems commonly referred to as Badhiriya (hearing loss) in ayurvedic terms. The traditional Ayurvedic medicine focuses on diet and natural herbs as a treatment solution to hearing loss, with the view that these herbs will have beneficial healing effects over the complete human body in a natural way, including the ear.<ref>{{Cite journal|last=Kotwal|first=S, Bisht K, Singh DS|date=2018|title=Hearing loss (Badhirya) and its Ayurvedic management: a case study|url=https://www.wisdomlib.org/uploads/journals/wjpr/volume-7,-december-issue-19_11483.pdf|journal=World Journal of Pharmaceutical Research|volume=7|pages=1319-1327}}</ref> <ref>{{Cite journal|last=Prajapati|first=Sweta R|last2=Joshi|first2=Shraddha|last3=Vaghela|first3=D B|date=2023-10-16|title=Effect of Vidaryadi Ghrita and Ksheerabala Oil in the management of hearing loss: a case series|url=https://www.joinsysmed.com/doi/10.4103/jism.jism_8_23|journal=Journal of Indian System of Medicine|language=en|volume=11|issue=3|pages=149–155|doi=10.4103/jism.jism_8_23|issn=2320-4419}}</ref> Typically, children born with significant hearing loss have been educated in special schools for hearing impairment, where education via Indian Sign Language is promoted.<ref>{{Cite journal|last=Jepson|first=Jill|date=1991-03|title=Urban and rural sign language in India|url=https://www.cambridge.org/core/product/identifier/S0047404500016067/type/journal_article|journal=Language in Society|language=en|volume=20|issue=1|pages=37–57|doi=10.1017/S0047404500016067|issn=0047-4045}}</ref> <ref>{{Cite journal|last=Zeshan|first=U, Vasishta MN, Sethna M|date=2005|title=Implementation of Indian Sign Language in educational settings|journal=Asia Pacific Disability Rehabilitation Journal|volume=16|pages=16-40}}</ref> There are two diploma courses: Diploma in Indian Sign Language Interpretation and Diploma in Teaching Indian Sign Language ([https://depwd.gov.in/ Department of Empowerment of Persons with Disabilities & Ministry of Social Justice and Empowerment], 2023). Audiological rehabilitation with amplification is becoming popular, and oral communication has become more prevalent. Many special schools have upgraded their mode of instruction to oral. Fully-fledged hearing services were established in some parts of the country as the field of audiology developed. Education of Audiology professionals in India first started at University level as a Masters program, similar to that in other countries. There are now a number of higher education institutions providing speech and hearing education and services across India. Also, hearing healthcare services are available throughout the country at different levels, although mainly in urban areas. === Education === India has robust training and education for audiologists compared to many low- and middle-income countries. In 1964, the first Audiology and Speech Language Therapy program was started at the twin institutes, BYL Nair Charitable Hospital and T. N. Medical College in Mumbai. In the same year, the All India Institute of Speech and Hearing (AIISH) was established by the Government of India, which is now a premier speech and hearing institute in Southeast Asia. Both programs were heavily influenced by American colleges and offered a dual degree in audiology and speech and language pathology. This dual degree practice is still present in most schools for bachelor's, however, the master's was bifurcated into masters in audiology and masters in speech language pathology From an educational qualification point of view, for an individual to work independently in India as an audiologist, the minimum qualification required is a Bachelor of Audiology and Speech Language Pathology (BASLP) from a recognized college acclaimed by the Rehabilitation Council of India (RCI). As per RCI norms, there are nearly 50 institutions that offer Bachelor’s degree and about 10 institutions that offer a Master’s degree in Speech and Hearing across the country. According to the [https://www.specialeducationnotes.in/2023/03/future-prospects-and-challenges-for.html Rehabilitation Council of India] (2023), approximately 750 candidates graduate at different levels each year. These programs are four years in duration and focus on speech and hearing sciences with approximately 1,500 hours of clinical practice. The typical program includes both audiology and speech language pathology, although various specialized master's programs in audiology and speech language pathology also exist ([https://hearinghealthmatters.org/hearing-international/2017/status-audiology-india/ Kumar Sanju et al.], 2017). To meet the demand, RCI initiated the Diploma Course in Hearing, Language and Speech (DHLS) to train Speech and Hearing Technicians for clinical assessment and therapeutic management of various speech, language, and hearing disorders. They work under guidance of fully-trained graduate or a postgraduate Speech & Hearing clinician and are given the designation as “Speech and Hearing Technicians" ([https://www.specialeducationnotes.in/2023/03/future-prospects-and-challenges-for.html Rehabilitation Council of India,] 2023). The Rehabilitation council of India (RCI) provides accreditation for these programs. The curriculum is regularly updated through RCI-mandated workshops and all the schools follow a minimum common curriculum. Many of these programs are internationally recognized, offering students the opportunity to study with people of different linguistic and cultural backgrounds. To assist in this, the course has an expectation that students be able to communicate in English. Following are some of the audiology programs in India: *Topiwala National Medical College c/o BYL Nair Charitable Hospital *All India Institute of Speech & Hearing (AIISH), Mysore University *Ali Yavar Jung Institute for the Hearing Handicapped (AYJNIHH) in four different cities across India *Manipal College of Allied Health Sciences, Manipal University *Sri Ramachandra Faculty of Audiology and Speech Language Pathology, *Sri Ramachandra Institute of Higher Education and Research (SRIHER) *Kasturba Medical College, Mangalore *Bharati Vidyapeeth Deemed University, Pune *MERF Institute of Speech and Hearing, Chennai *Post Graduate Institute of Medical Education & Research (PGIMER) The list of RCI-approved institutions are available at [https://www.ishaindia.org.in/find_a_college Indian Speech and Hearing Association] Following are some of the courses: *Bachelors in Audiology and Speech Language Pathology (BASLP) *Bachelor of Education – Hearing Impaired (BEd-HI) *Masters of Science – Audiology and Speech Language Pathology (MSc ASLP) *Masters of Science – Audiology *Masters of Science – Speech Language Pathology (MSc SLP) *Master of Education – Hearing Impaired (MEd – HI) *Doctor of Philosophy (PhD) – Audiology *Doctor of Philosophy (PhD) – Speech Language Pathology *Doctor of Philosophy (PhD) – Speech and Hearing Sciences === Audiological Services === Audiological services in India include hearing assessment, selection and fitting of hearing aids, and aural rehabilitation. Some of the centers have successful cochlear implantation programs; however, the services in some of the specialized audiology areas like vestibular assessment and rehabilitation, assessment and management of auditory processing disorders, and tinnitus rehabilitation are limited. It is important to note that most audiological facilities are based in urban areas, making it difficult for people in rural areas to access such services. A few public sector organizations and non-governmental organizations (NGOs) work to extend audiological services to rural and remote areas by conducting residential camps and appointing public health workers to facilitate identification of hearing disorders and appropriate referral. Hence, although there is a great need, demand for audiological services in rural areas is limited, and patients generally tend to travel to urban areas to use the available services. Services offered in the public sector are either paid for or subsidized by the government. However, the patients generally pay for services offered by the private facilities, although in some instances various NGOs and charities may pay for them, especially for children. Audiologists in the private sector are greater in number than those in the public sector. These are generally equipped with all the necessary diagnostic instruments and their work is mainly focused in hearing aid dispensing. The patient has to pay for private sector service. There are also some well-known private cochlear implant centers across South Asia that attract patients from other countries. Generally, these clinics or institutes are not easily accessible to people living in rural settings, as most of them are based in urban locations. In addition, a concern in relation to private sector provision is that the practice of hearing aid dispensing is not well regulated, although some regulations from the Rehabilitation Council of India (RCI) exist. Public sector facilities with audiology services are mostly district-level hospitals, educational institutes, and district differently abled welfare offices. The services provided are free or at concessionary rates. === Professionals === Generally, audiologists and otolaryngologists mainly offer specialist audiological services in India. A few otologists and otoneurologists practice exclusively in their specialty. However, various professionals, including general medical practitioners, teachers of the deaf, health workers, and community volunteers at various levels, offer some of the services. There are more than 6,000 registered Audiology and Speech-Language Pathology professionals in India.<ref>{{Cite journal|last=Nayaka|first=S. Hemaraja|last2=Subramaniam|first2=Vijayalakshmi|date=2021-01|title=Journey of Hearing Health Care in India: Historical Perspectives|url=https://journals.lww.com/10.4103/amhs.amhs_125_21|journal=Archives of Medicine and Health Sciences|language=en|volume=9|issue=1|pages=151–155|doi=10.4103/amhs.amhs_125_21|issn=2321-4848}}</ref> In 2021 it was estimated that there is one audiologist for every 500 000 individuals in India, while the number of otorhinolaryngologists is one per 140 0006,55 (the WHO recommendation is one per 25,00054). In India, most of the audiologists are located in the southern part of the country. This skewed distribution of professionals is due to the geographical locations of institutes offering bachelor and master degree programs in speech and hearing. In the northern part of India (the states of Jammu and Kashmir, Delhi, Punjab, Rajasthan, Chandigarh, Uttarakhand, Himachal Pradesh, Uttar Pradesh, and Bihar), there are only 7 institutions that offer the BASLP (Bachelor of Audiology and Speech-Language Pathology degrees) and only one institute (PGI Chandigarh that has a yearly student intake capacity of 2) that offers a MASLP program. There is considerable brain drain with westward migration of audiologists due to low salaries.<ref>{{Cite journal|last=Singh|first=Niraj Kumar|last2=Rao|first2=Amulya P.|last3=Krishna|first3=Y.|last4=Arun|first4=B.|last5=Yathiraj|first5=Asha|last6=Indranil|first6=C.|last7=Sunil|first7=K. R|last8=Pradeep|last9=Kumar|first9=Prawin|date=2022-01|title=Factors Leading to Brain Drain of Speech and Hearing Professionals in India|url=https://journals.lww.com/10.4103/jisha.jisha_25_21|journal=Journal of Indian Speech Language & Hearing Association|language=en|volume=36|issue=1|pages=25–30|doi=10.4103/jisha.jisha_25_21|issn=0974-2131}}</ref> The majority are employed in India; however, many of them have found employment in the USA, Australia, the UK, New Zealand, and the Gulf countries. In recent years, there has been an increase in the global demand for audiologists due to the modernization of audiology, especially in western countries. This has resulted in a major drain of skilled audiologists to western countries and a shortage of qualified audiology professionals in India. === Professional and Regulatory Bodies === The [https://www.ishaindia.org.in/ Indian Speech and Hearing Association] (ISHA) is the professional and scientific association with over 2,500 members, while the Rehabilitation Council of India (RCI) is the regulatory body that regulates training and professional practice. The Indian Speech and Hearing Association (ISHA) was formed in 1967. It is a professional and scientific association for audiologists and speech and language pathologists in India with over 2,500 registered members. Its role is the promotion of excellence in the speech, language, and hearing profession and rehabilitation services through advocacy, leadership, and continued education. It is also working to develop an ethical framework, monitor professionals, encourage networking, and support research. The Indian Speech and Hearing Association has provided a Scope of Practice document to guide the professionals and is available [https://www.ishaindia.org.in/public/PDF/Scope_of_Practice.pdf here]. The Rehabilitation Council of India (RCI) was set up as a registered society in 1986. In 1992, the Government of India implemented action to regulate the curriculum, training, and practice of rehabilitation courses under the Rehabilitation Council of India Act.<ref>{{Cite journal|last=Manchaiah|first=VKC, Sivaprasad MR, Chundu S|date=November 2009|title=Audiology in India|url=https://research.ebsco.com/c/cpafum/search/details/xu4zzl2vaf?db=a9h%2Cawn%2Cbth%2Ccin20%2Cnlebk%2Cecn%2Cega%2C8gh%2Clls%2Cnts%2Cbwh%2Cnsm&limiters=&q=IS%201535-2609%20AND%20VI%2021%20AND%20IP%206%20AND%20DT%202009|journal=Audiology Today|volume=21|pages=38-44}}</ref> Apart from monitoring the curriculum, RCI has also laid down strict norms for practicing rehabilitation sciences. RCI also maintains a Central Rehabilitation Register (CRR) of all qualified professionals and personnel working in the field of rehabilitation and special education, which requires registration and periodic renewal. The RCI act mandates membership of CRR for practicing allied health professionals. The RCI also prescribes disciplinary action against unqualified persons delivering services to persons with disability, although such efforts have been limited. === Workforce Problem === India's speech and hearing professionals predominantly reside in the southern region due to the geographical distribution of institutes offering these degrees. In the northern region, only seven institutions offer BASLP degrees, and one institute offers MSc programs. Certificate holders (DIPLOMA/ audiometricians) are being hired and expected to work without supervision instead of audiologists in various private and public sectors as salaries can be further reduced. This situation exists in both public and private health systems, and there is migration of these diploma holders to major cities. As a result, the rural areas, tier two, and three cities that were supposed to receive supervised services from diploma holders do not have any services. Further, private hearing care clinics, as well as hearing aid and cochlear implant corporations, provide low salaries. This is challenging for audiologists with graduate and postgraduate degrees, and the greater compensation offered by hearing care clinics in countries such as the United States, Australia, and the United Kingdom encourages many competent audiologists to leave India and work elsewhere. Another challenge in the delivery of hearing care is that there are more audiologists in the private sector than the public sector. === State of Ear and Hearing Care - Policy === I. National strategy implemented through Ministry of Health as part of National Health Mission; * National Programme for Prevention and Control of Deafness (Ministry of Health and Family Welfare, 2016) for preventable hearing loss- across age groups and focused on preventable/acquired hearing losses. * Rashtriya Bal Swasthya Karyakram is implemented through the District General Hospital (Rashtriya Bal Swasthya Karyakram, 2013) and early diagnostic and intervention centres for childhood disability, including hearing loss. II. National strategy implemented through Ministry of Social Justice and Empowerment *Screening camps through Ministry of education (SSA) and through District Differently Abled welfare office *Provision of hearing aids through ADIP scheme *Provision of disability ID cards *Allowances for persons with disabilities III. State-level strategies: *Provision of cochlear implants through Chief minister schemes *Allowances for persons with disabilities. State of reimbursement of audiology services and aural rehabilitation services (Archana et al., 2016) *National health plan: Only hearing aids are covered under the CGHS scheme (for central government employees and their family members). (Ministry of Health and Family Welfare, 2020) *National level-speech therapy: CGHS issued guidelines for reimbursement of Speech Therapy, Occupational Therapy, Behavioral Therapy for Children with ASD, ADHD and SLI. Reimbursement Ceiling Rate Per Session: Rs.400 (Controller General of Defense Accounts, 2023) *State level: Cochlear implant surgeries, AVT (for 1 year), accessories are covered in insurance (Sharma et al., 2024) *State level: Tamil Nadu state is considering a rights-based social model with inclusion of speech therapy services among many other therapy services under insurance. {{HTitle|Research in Audiology}} Research in Audiology is predominantly from educational institutions. In the formative years of the profession, audiology research was to develop suitable locally relevant materials for testing speech identification, or speech perception. Adapting western tools and validating these tools were some of the lines of work. Parallelly, some institutions pursued basic science research in hearing, while other institutions gravitated to clinical and translational research. Some institutions have received extramural funding to pursue research in specific areas such as speech perception in relation to hearing aid technology and vestibular research at AIISH, Mysuru, community-based participatory models for hearing care using e and mhealth at SRIHER (DU), Chennai; research on perception of spatial cues, spatial release from masking, and vestibular research at KMC Mangalore, Tinnitus research at the School of Allied Health Sciences, MAHE, Manipal; and vestibular research at Bharathi Vidya Peeth, Pune. The scientific community over the years has diversified its publication from national to international peer-reviewed journals in the past decade. The Indian Speech and Hearing Association has its own journal called the Journal of Indian Speech and Hearing Association, where predominantly postgraduate student dissertations or student research papers are published. There is also the Journal of All India Institute of Speech and Hearing, which represents work from AIISH, Mysuru and some other Indian studies. Despite this, research gaps exist in understanding some pertinent epidemiological evidence of hearing loss across age groups and multi-centric cohort studies to build robust evidence on outcomes of early identification and cochlear implantation or hearing aids. Also, more cohort case-control studies to establish more valid outcomes of clinical relevance may have to be considered to advance hearing care. {{HTitle|Audiology Charities}} In recent years, many non-governmental organizations (NGOs) and charities have become very active and are working towards improving ear and hearing healthcare services or facilities in India. The following are some of the major non-governmental organizations or charities: * [https://www.audiologyindia.org/ Audiology India] * [https://aured.org/ Aural Education for the Hearing Impaired] * [https://www.deedsindia.org/ Development Education Empowerment for the Disadvantaged in Society] (DEEDS) * [http://www.ihearfoundation.org/ I Hear Foundation] * [https://en.wikipedia.org/wiki/Nambikkai Nambikkai Foundation] * [http://www.readsindia.org/ Research Education & Audiological Development Society] (READS) * [http://www.rotaryfoundationindia.org/ Rotary Foundation (India)] {{HTitle|Challenges, Opportunities and Notes}} ===Challenges === *Awareness and access to hearing health are still major concerns in the rural population, where the majority of the Indian population lives. Hence, there is a need to adopt a public health approach and community-based hearing rehabilitation. The complexity in terms of educational, religious, and socioeconomic backgrounds of such a diverse population needs to be considered in this. *Health literacy, superstitions, finances, and local access to services are the major barriers to hearing healthcare. *A developing middle-class (middle-income) population has created a new demand for hearing healthcare services. *Ensuring an even geographical distribution of audiology professionals and infrastructure and improving accessibility to audiological services for people living in remote and rural settings. *There is a great need for developing training and clinical services in areas including auditory processing disorders, vestibular disorders, and tinnitus. *There is a need to better define the scope of practice for audiologists with different training levels and to develop standardized procedures for practice, which may result in more uniform service provision. *The Defense Research and Development Organization (DDRO) is working towards the development of an indigenous cochlear implant. This could significantly bring the cost down, making it more affordable for low and middle-income families and helping over a million children who suffer from profound hearing impairment. *Although private sector practices have state-of-the-art facilities, the practice is not well regulated, resulting in many unqualified (or poorly trained) individuals practicing. Hence, much effort is needed from RCI and ISHA to enforce the necessary practice regulations. *Audiology practice in India is based on models from western countries. Considering that social and cultural aspects vary vastly in India compared to western countries, there is a great need to develop research in India that should inform practice. Hence, there is a need for improving clinical and applied research, initially starting with epidemiological studies to better understand the extent and nature of hearing disorders. *Many charities and NGOs have been working actively to improve hearing healthcare services, especially in rural areas. *There is a need for the establishment of patient organizations, which may provide a platform for people with hearing impairment and their family members to share ideas and concerns to better promote hearing healthcare. *Reducing the brain drain and increasing the manpower of hearing healthcare professionals *Raise funding for both clinical and research work through the government and, various national and international charities and organizations ===SWOT or SCORE analysis of the country situation=== No SWOT analysis conducted for rehabilitation professionals -audiologist / ear and hearing care providers. Only one SWOT addressed Allied Health Professionals but that is restricted to the context training from national institutions such as the [https://en.wikipedia.org/wiki/All_India_Institutes_of_Medical_Sciences All India Institutes of Medical Science] and the [https://en.wikipedia.org/wiki/Postgraduate_Institute_of_Medical_Education_and_Research Postgraduate Institute of Medical Education and Research] (PGI). <ref>{{Cite journal|last=Chaudhary|first=P|date=2018|title=The Status of Allied Health Professionals in India: Need for a SWOT analysis|journal=Amity Journal of Healthcare Management|volume=2018|pages=3-9}}</ref> World Health Organization tools were not used for this analysis. This analysis which is un-specific to audiology but yet is somewhat common is as follows: '''Strengths''' * Institutions of good quality and reputation (nationally & internationally) * High quality Laboratories * Tradition and knowledge transfer * Research & Development on priority basis * Exposure to high profile professionals and peers * Exposure to special expertise and high-end technology * Low cost but strong infrastructure * High quality results * Competent workforce * Certain rehabilitation professions are Governed by a council '''Weaknesses''' * Some rehabilitation professions are not governed by council * Inability to meet demand * Uneven geographical distribution * Low paid job opportunities * Limited utilization of AHPs * Lack of associations and union activities * Low priority areas for the GOI * Limited career options * Very weak promotion avenues after employment * Social stigma due to low esteem resulting into high rate of brain drain * No dedicated infrastructure for training of AHPs * No dedicated faculty (doctors are acting as teachers who are already overburdened) * No attention is paid to faculty development program * Lack of motivation and recognition '''Opportunities''' * Arrival of new medical technologies. * Emergence of new marketing opportunities. * Advancement in technology demand trained individuals who can handle sophisticated machinery to produce reliable results in conjunction with patient safety. * Health sector reform at national level. * Today, there is an urgent need for competent people for accreditation and licensing of healthcare organizations. * Better job prospects both globally and nationally (Job Outlook, 2018) '''Threats''' * Ever changing technology; (Evolving technologies you’re unprepared for) * Changing market trends * New and increased competition * Economic slowdowns/ difficulties * Lack of standard protocols for their education and practical training in different parts of India * Mushroom growth of unauthorized teaching institutes; giving diplomas/ degrees without providing quality teaching or practical training ===Summary of Gaps=== # '''Educational Quality:''' Ensuring that RCI-approved institutions and professionals provide high-quality education and training is an enormous challenge. Advocacy is not strongly promoted in the curriculum or training # '''Lack of mainstreaming of rehabilitation:''' Many people in India remain unaware of the purpose and significance of rehabilitation services. This is because rehabilitation is not mainstream in the health system. # '''Skilled Professional Shortage:''' Skilled professionals are in short supply in India's rehabilitation sector. The RCI must address this issue by encouraging and regulating the training and education of experts in this field. # '''Decentralized health-care facilities and facility centers''' using the Public-Private Partnership Model and competent institutions offering knowledge and services can address the challenges.(Nayaka & Subramaniam, 2021) # '''Increasing Demand for Rehabilitation Services:''' With an aging population and increasing prevalence of disabilities due to chronic diseases and accidents, there is a growing demand for rehabilitation services. # '''Technological Advancements:''' The use of technology in rehabilitation services is on the rise. The RCI can leverage this trend to enhance training and education for professionals and ensure that they are equipped with the latest knowledge and skills. # '''Government Support:''' Increasing salaries for audiologists, increasing number of positions within the public health system can increase the reach of services.(D.ED SPECIAL EDUCATION, 2023) # '''Information that could have been used to help identify problems and resources:''' Publications on alternative models of care / implementation published from India could have served to guide programs like RBSK , NPPCS and other govt programs. {{HTitle|References}} {{reflist}} {{Global Audiology Authors |name1=Vidya Ramkumar |role1=Author |website1=https://sites.google.com/sriramachandra.edu.in/sresht-sriher-ia-slhs-lab/team/dr-vidya-ramkumar-cv |name2=Deeptaa Prabhakar |role2=Author |name3=R. Vishnu Saravana |role3=Author |linkedin3=https://in.linkedin.com/in/vishnu-saravana-a25974198?trk=public_profile_browsemap_profile-result-card_result-card_full-click |name4=Vinaya Manchaiah |role4=Author |website4=https://www.vinayamanchaiah.com/ }} [[Category:India]] [[Category:Audiology]] mktmzrp25nbmcwaqq4tjvy6esyz2ahf 0 296049 2806737 2805024 2026-04-27T03:29:46Z RadiX 155307 RadiX moved page [[Global Audiology/Asia/Hong Kong SAR]] to [[Global Audiology/Asia/Hong Kong]] without leaving a redirect: fix 2805024 wikitext text/x-wiki {{:Global Audiology/Header}}{{:Global Audiology/Asia/Header}}{{CountryHeader|File:Hong Kong on the globe (Southeast Asia centered).svg|https://en.wikipedia.org/wiki/Hong_Kong}}{{HTitle|Brief Country Information }} [https://en.wikipedia.org/wiki/Hong_Kong Hong Kong] is a special administrative region of China. Situated on China's southern coast just south of Shenzhen, it consists of Hong Kong Island, Kowloon, and the New Territories. Chinese and English are the official languages of Hong Kong. English is widely used in the Government and by the legal, professional, and business sectors. Trilingual professionals who speak English, Cantonese, and Putonghua (Mandarin Chinese) play a vital role in the numerous professional fields in Hong Kong. {{HTitle|Incidence and Prevalence of Hearing Loss }}A territory-wide survey on persons with disabilities and chronic diseases was conducted during August 2019 - December 2020 via the local continuous sample survey, the General Household Survey, to estimate the total number and prevalence rate of persons with selected types of disabilities and chronic diseases. In 2020, some 266,900 people (or 3.6% of the total population) in Hong Kong reported that they had some difficulty in hearing, 44,300 people (0.6%) had a lot of difficulty, and 3,600 people (less than 0.05%) could not hear at all. Among the total population, some 53,400 people (0.7%) reported using a hearing aid or tool. Among the 266,900 people who reported that they had some difficulty in hearing, 9.4% cited that they were using a hearing aid or cochlear implant. Among the 44,300 people who had a lot of difficulty in hearing, 24.0% cited that they were using a hearing aid or cochlear implant. And among the 3,600 people who could not hear at all, 9.3% cited that they were using a hearing aid or cochlear implant. Of the 47,900 people aged 2 and over with hearing difficulty, 3,000 (6.3%) reported that they use sign language in their usual communication. In Hong Kong, disability statistics are mainly available from relevant Government bureaus or departments, statutory bodies, and nongovernmental organizations. In particular, the Central Registry for Rehabilitation of the Labour and Welfare Bureau issues the Registration Card for People with Disabilities (Registration Card) to persons with permanent or temporary disabilities as a documentary proof of their disability status and maintains relevant records. Persons with disabilities and with certification by registered medical practitioners or allied health professionals, etc., could apply for the Registration Card. There were some 93,000 holders of valid Registration cards as of March 2021. As such applications are on a voluntary basis, it is assumed that such figures are some sort of lower bound estimates of the number of people with the specific type of disability in Hong Kong.² {{HTitle|Information About Audiology}} === Educational Institution === The Faculty of Education at the University of Hong Kong was one of the first in Southeast Asia and China to offer post-graduate training in audiology. The first students were enrolled in 1996, with intakes every two years. The two-year program provides students with detailed theoretical background knowledge of human hearing and hearing loss, as well as intensive clinical practice in a variety of audiology placements. The medium of instruction is English. The program is designed to provide students with high-quality clinical skills and an appreciation of research in audiology.³ === Professional Bodies === ==== Hong Kong Society of Audiology ==== The Hong Kong Society of Audiology is a voluntary organization founded in 1992 by a group of audiologists in Hong Kong. The Society has grown steadily over the years. In 2023, The Society has approximately 120 professional members.⁴ The Hong Kong Society of Audiology Limited endeavours: * To share among members the information and technology in Audiology and other related fields so as to update and promote the standard of audiological assessments and rehabilitation services in Hong Kong. * To promote research in the area of Audiology and related fields with reference * To local needs for study, diagnosis, alleviation, and prevention of hearing impairment. * To serve as a channel of communication among members and local and overseas professional bodies in matters related to Audiology, education, hearing, and speech sciences. * To serve as a consultative body for other professional organisations and community bodies on matters related to Audiology education, hearing, and speech sciences. * To serve as a social group for members of the Society and to promote their welfare in works related to Audiology, education, hearing, and speech sciences. ==== Hong Kong Institute of Audiologists ==== The Hong Kong Institute of Audiologists (HKIA) was established in March 2018 to enhance the professional standard of audiologists in Hong Kong and to participate in the Accredited Registers Scheme for Healthcare Professions administered by the Government of the Hong Kong Special Administrative Region (6). HKIA ensures that the local public has access to professional audiological assessment and rehabilitation services provided by qualified audiologists. The Institute has approximately 60 audiologist registrants in 2023.⁵ The missions of the HKIA include: * To maintain professional standards of audiology services in Hong Kong; * To safeguard the public's interests in accessing audiology services in Hong Kong; * To maintain adequate standards of professional practice for audiologists; * To promote adequate standards of professional practice and of professional conduct among audiologist registrants; * To establish and maintain the accredited register of audiologists in Hong Kong; * To establish and maintain contact with other members of the audiology profession in Hong Kong. {{HTitle|Research in Audiology}}The majority of audiology research in Hong Kong is led by research teams from The University of Hong Kong, Education University of Hong Kong, the Chinese University of Hong Kong. {{HTitle|Audiology practice}} ==== Elderly Health Care Voucher Scheme ==== The coverage of the Elderly Health Care Voucher Scheme (EHVS) in Hong Kong was extended to include primary healthcare services provided by four categories of the healthcare profession under the Accredited Registers Scheme for Healthcare Professions (i.e., audiologists, dietitians, clinical psychologists, and speech therapists), as well as medical equipment (such as hearing aids) provided by them upon professional assessment in April 2023. It is estimated that almost 1.7 million eligible elderly people in Hong Kong benefit from the EHVS. Together with the four new categories of healthcare professions, eligible elderly persons are able to make use of vouchers to pay for primary healthcare services provided by a total of 14 categories of healthcare professions (i.e., medical practitioners, Chinese medicine practitioners, dentists, nurses, physiotherapists, occupational therapists, radiographers, medical laboratory technologists, chiropractors, optometrists with Part I registration, audiologists, dietitians, clinical psychologists, and speech therapists). As of March 2023, over 11,000 healthcare service providers in Hong Kong had already enrolled in the EHVS. Eligible elderly persons can use vouchers at over 33,000 service points across the territory. Under the EHVS, eligible persons aged 65 and above are given an annual voucher amount to pay for services provided by private primary healthcare service providers enrolled in the EHVS. Any unused voucher amount can be carried forward for use in the following years, subject to a maximum accumulation limit.⁷ {{HTitle|Audiology Charities}} ==== The Hong Kong Society for the Deaf ==== Found in 1968, the Hong Kong Society for the Deaf is a non-profit making organisation which aims to promote the well-being of the hearing impaired and seeks to ensure equalisation of opportunities for hearing impaired persons. The Society aims to provide comprehensive and professional services of the highest standards to ensure equalisation of opportunities for the hearing impaired, and to promote self-development, self-actualisation and self-sufficiency among the hearing impaired so they could integrate into society. Its objectives include: * To undertake projects of publicity, education, recreation, counseling, audiological and medical services for the hearing impaired; and to assist or collaborate with any institutions, organisations or individuals to improve the services for the hearing impaired. * To work towards improving the educational standards for the hearing impaired, to provide scholarships and special equipment for the hearing impaired, and to provide guidance for parents of hearing impaired children. * To inform the public about the problems and needs of the hearing impaired, to give necessary information to hearing impaired persons and their families about institutions and services available to them, and to exchange information among institutions serving the hearing impaired and to work towards the integration of the hearing impaired with the general public.⁸ ==== Hear Talk Foundation ==== Launched in 2003, Hear Talk Foundation is a registered charitable organisation in Hong Kong committed to serving the underprivileged communities with hearing impairment and speech disorders both in Hong Kong and Mainland China, especially children and the elderly. It has been established by a group of committed ENT specialists, audiologists, speech therapists, and educators.⁹ {{HTitle|Challenges, Opportunities and Notes}} === Challenges === * There is a shortage of manpower in audiology, both in clinical practices and in the academic field. * Health literacy and cultural factors are the main barriers to the utilization of hearing health services. * The majority of the local population speaks Cantonese Chinese as their native language. Only a very limited number of speech assessment materials (including speech recognition tests, speech perception tests, and speech audiometry) are available in the local language. * There is no regulation for any hearing aid or hearing device. There is also no statutory registration for anyone practicing in the audiology field. {{HTitle|References}} {{reflist}} [[Category:Audiology]] [[Category:Hong Kong]] <ref>{{Cite web|url=https://www.gov.hk/en/about/abouthk/facts.htm|title=GovHK: Hong Kong – the Facts|last=GovHK (www.gov.hk)|website=www.gov.hk|language=en|access-date=2023-06-26}}</ref><ref>Census and Statistics Department, Hong Kong SAR. Social data collected via the General Household Survey Special Topics Report No. 63: Persons with disabilities and chronic diseases. 2021; Available at: <nowiki>https://www.censtatd.gov.hk/en/data/stat_report/product/C0000055/att/B11301632021XXXXB0100.pdf</nowiki>. Accessed Jun 21, 2023.</ref><ref>Faculty of Education, The University of Hong Kong. Master of Science in Audiology. 2023; Available at: <nowiki>https://web.edu.hku.hk/programme/audiology</nowiki>. Accessed Jun 21, 2023.</ref><ref>{{Cite web|url=https://www.audiology.org.hk/about/our-mission/|title=HKSA and Our Mission – Hong Kong Society of Audiology|language=en-GB|access-date=2023-06-26}}</ref><ref>{{Cite web|url=https://www.audiologists.org.hk/about-hkia-2/|title=Audiologists 聽力學家 – Hong Kong Institute of Audiologists|language=en-GB|access-date=2023-06-26}}</ref><ref>{{Cite web|url=https://www.ars.gov.hk/en/accr_pro_bodies.html|title=Accredited Registers Scheme for Healthcare Professions - Accredited Healthcare Professional Bodies|website=www.ars.gov.hk|access-date=2023-06-26}}</ref><ref>Press Releases, The Government of the Hong Kong SAR. Coverage of Elderly Health Care Voucher Scheme to extend to include four categories of healthcare profession under Accredited Registers Scheme for Healthcare Professions. 2023; Available at: <nowiki>https://www.info.gov.hk/gia/general/202304/27/P2023042700410.htm</nowiki>. Accessed Jun 21, 2023.</ref><ref>{{Cite web|url=https://www.deaf.org.hk/en/mission.php|title=The Hong Kong Society for the Deaf|website=www.deaf.org.hk|access-date=2023-06-26}}</ref><ref>{{Cite web|url=http://www.heartalk.org/en/about_us/our_mission/|title=Our Mission {{!}} Hear Talk Foundation|website=www.heartalk.org|access-date=2023-06-26}}</ref> <references /> {{:Global Audiology/Authors-1|NG Hoi Yee Iris|http://www.ihcr.cuhk.edu.hk/professor-iris-hoi-yee-ng/}} ''Edited in part by'' [http://linkedin.com/in/joyce-rodvie-sagun-4691bb182 Joyce Rodvie Sagun] m0ntzw92md8i2e1zw5xwhieoaw9acw3 2806762 2806737 2026-04-27T04:33:02Z RadiX 155307 upd 2806762 wikitext text/x-wiki {{:Global Audiology/Header}}{{:Global Audiology/Asia/Header}}{{CountryHeader|File:Hong Kong on the globe (Southeast Asia centered).svg|https://en.wikipedia.org/wiki/Hong_Kong}}{{HTitle|Brief Country Information }} [https://en.wikipedia.org/wiki/Hong_Kong Hong Kong] is a special administrative region of China. Situated on China's southern coast just south of Shenzhen, it consists of Hong Kong Island, Kowloon, and the New Territories. Chinese and English are the official languages of Hong Kong. English is widely used in the Government and by the legal, professional, and business sectors. Trilingual professionals who speak English, Cantonese, and Putonghua (Mandarin Chinese) play a vital role in the numerous professional fields in Hong Kong. {{HTitle|Incidence and Prevalence of Hearing Loss }}A territory-wide survey on persons with disabilities and chronic diseases was conducted during August 2019 - December 2020 via the local continuous sample survey, the General Household Survey, to estimate the total number and prevalence rate of persons with selected types of disabilities and chronic diseases. In 2020, some 266,900 people (or 3.6% of the total population) in Hong Kong reported that they had some difficulty in hearing, 44,300 people (0.6%) had a lot of difficulty, and 3,600 people (less than 0.05%) could not hear at all. Among the total population, some 53,400 people (0.7%) reported using a hearing aid or tool. Among the 266,900 people who reported that they had some difficulty in hearing, 9.4% cited that they were using a hearing aid or cochlear implant. Among the 44,300 people who had a lot of difficulty in hearing, 24.0% cited that they were using a hearing aid or cochlear implant. And among the 3,600 people who could not hear at all, 9.3% cited that they were using a hearing aid or cochlear implant. Of the 47,900 people aged 2 and over with hearing difficulty, 3,000 (6.3%) reported that they use sign language in their usual communication. In Hong Kong, disability statistics are mainly available from relevant Government bureaus or departments, statutory bodies, and nongovernmental organizations. In particular, the Central Registry for Rehabilitation of the Labour and Welfare Bureau issues the Registration Card for People with Disabilities (Registration Card) to persons with permanent or temporary disabilities as a documentary proof of their disability status and maintains relevant records. Persons with disabilities and with certification by registered medical practitioners or allied health professionals, etc., could apply for the Registration Card. There were some 93,000 holders of valid Registration cards as of March 2021. As such applications are on a voluntary basis, it is assumed that such figures are some sort of lower bound estimates of the number of people with the specific type of disability in Hong Kong.² {{HTitle|Information About Audiology}} === Educational Institution === The Faculty of Education at the University of Hong Kong was one of the first in Southeast Asia and China to offer post-graduate training in audiology. The first students were enrolled in 1996, with intakes every two years. The two-year program provides students with detailed theoretical background knowledge of human hearing and hearing loss, as well as intensive clinical practice in a variety of audiology placements. The medium of instruction is English. The program is designed to provide students with high-quality clinical skills and an appreciation of research in audiology.³ === Professional Bodies === ==== Hong Kong Society of Audiology ==== The Hong Kong Society of Audiology is a voluntary organization founded in 1992 by a group of audiologists in Hong Kong. The Society has grown steadily over the years. In 2023, The Society has approximately 120 professional members.⁴ The Hong Kong Society of Audiology Limited endeavours: * To share among members the information and technology in Audiology and other related fields so as to update and promote the standard of audiological assessments and rehabilitation services in Hong Kong. * To promote research in the area of Audiology and related fields with reference * To local needs for study, diagnosis, alleviation, and prevention of hearing impairment. * To serve as a channel of communication among members and local and overseas professional bodies in matters related to Audiology, education, hearing, and speech sciences. * To serve as a consultative body for other professional organisations and community bodies on matters related to Audiology education, hearing, and speech sciences. * To serve as a social group for members of the Society and to promote their welfare in works related to Audiology, education, hearing, and speech sciences. ==== Hong Kong Institute of Audiologists ==== The Hong Kong Institute of Audiologists (HKIA) was established in March 2018 to enhance the professional standard of audiologists in Hong Kong and to participate in the Accredited Registers Scheme for Healthcare Professions administered by the Government of the Hong Kong Special Administrative Region (6). HKIA ensures that the local public has access to professional audiological assessment and rehabilitation services provided by qualified audiologists. The Institute has approximately 60 audiologist registrants in 2023.⁵ The missions of the HKIA include: * To maintain professional standards of audiology services in Hong Kong; * To safeguard the public's interests in accessing audiology services in Hong Kong; * To maintain adequate standards of professional practice for audiologists; * To promote adequate standards of professional practice and of professional conduct among audiologist registrants; * To establish and maintain the accredited register of audiologists in Hong Kong; * To establish and maintain contact with other members of the audiology profession in Hong Kong. {{HTitle|Research in Audiology}}The majority of audiology research in Hong Kong is led by research teams from The University of Hong Kong, Education University of Hong Kong, the Chinese University of Hong Kong. {{HTitle|Audiology practice}} ==== Elderly Health Care Voucher Scheme ==== The coverage of the Elderly Health Care Voucher Scheme (EHVS) in Hong Kong was extended to include primary healthcare services provided by four categories of the healthcare profession under the Accredited Registers Scheme for Healthcare Professions (i.e., audiologists, dietitians, clinical psychologists, and speech therapists), as well as medical equipment (such as hearing aids) provided by them upon professional assessment in April 2023. It is estimated that almost 1.7 million eligible elderly people in Hong Kong benefit from the EHVS. Together with the four new categories of healthcare professions, eligible elderly persons are able to make use of vouchers to pay for primary healthcare services provided by a total of 14 categories of healthcare professions (i.e., medical practitioners, Chinese medicine practitioners, dentists, nurses, physiotherapists, occupational therapists, radiographers, medical laboratory technologists, chiropractors, optometrists with Part I registration, audiologists, dietitians, clinical psychologists, and speech therapists). As of March 2023, over 11,000 healthcare service providers in Hong Kong had already enrolled in the EHVS. Eligible elderly persons can use vouchers at over 33,000 service points across the territory. Under the EHVS, eligible persons aged 65 and above are given an annual voucher amount to pay for services provided by private primary healthcare service providers enrolled in the EHVS. Any unused voucher amount can be carried forward for use in the following years, subject to a maximum accumulation limit.⁷ {{HTitle|Audiology Charities}} ==== The Hong Kong Society for the Deaf ==== Found in 1968, the Hong Kong Society for the Deaf is a non-profit making organisation which aims to promote the well-being of the hearing impaired and seeks to ensure equalisation of opportunities for hearing impaired persons. The Society aims to provide comprehensive and professional services of the highest standards to ensure equalisation of opportunities for the hearing impaired, and to promote self-development, self-actualisation and self-sufficiency among the hearing impaired so they could integrate into society. Its objectives include: * To undertake projects of publicity, education, recreation, counseling, audiological and medical services for the hearing impaired; and to assist or collaborate with any institutions, organisations or individuals to improve the services for the hearing impaired. * To work towards improving the educational standards for the hearing impaired, to provide scholarships and special equipment for the hearing impaired, and to provide guidance for parents of hearing impaired children. * To inform the public about the problems and needs of the hearing impaired, to give necessary information to hearing impaired persons and their families about institutions and services available to them, and to exchange information among institutions serving the hearing impaired and to work towards the integration of the hearing impaired with the general public.⁸ ==== Hear Talk Foundation ==== Launched in 2003, Hear Talk Foundation is a registered charitable organisation in Hong Kong committed to serving the underprivileged communities with hearing impairment and speech disorders both in Hong Kong and Mainland China, especially children and the elderly. It has been established by a group of committed ENT specialists, audiologists, speech therapists, and educators.⁹ {{HTitle|Challenges, Opportunities and Notes}} === Challenges === * There is a shortage of manpower in audiology, both in clinical practices and in the academic field. * Health literacy and cultural factors are the main barriers to the utilization of hearing health services. * The majority of the local population speaks Cantonese Chinese as their native language. Only a very limited number of speech assessment materials (including speech recognition tests, speech perception tests, and speech audiometry) are available in the local language. * There is no regulation for any hearing aid or hearing device. There is also no statutory registration for anyone practicing in the audiology field. {{HTitle|References}} {{reflist}} [[Category:Audiology]] [[Category:Hong Kong]] <ref>{{Cite web|url=https://www.gov.hk/en/about/abouthk/facts.htm|title=GovHK: Hong Kong – the Facts|last=GovHK (www.gov.hk)|website=www.gov.hk|language=en|access-date=2023-06-26}}</ref><ref>Census and Statistics Department, Hong Kong SAR. Social data collected via the General Household Survey Special Topics Report No. 63: Persons with disabilities and chronic diseases. 2021; Available at: <nowiki>https://www.censtatd.gov.hk/en/data/stat_report/product/C0000055/att/B11301632021XXXXB0100.pdf</nowiki>. Accessed Jun 21, 2023.</ref><ref>Faculty of Education, The University of Hong Kong. Master of Science in Audiology. 2023; Available at: <nowiki>https://web.edu.hku.hk/programme/audiology</nowiki>. Accessed Jun 21, 2023.</ref><ref>{{Cite web|url=https://www.audiology.org.hk/about/our-mission/|title=HKSA and Our Mission – Hong Kong Society of Audiology|language=en-GB|access-date=2023-06-26}}</ref><ref>{{Cite web|url=https://www.audiologists.org.hk/about-hkia-2/|title=Audiologists 聽力學家 – Hong Kong Institute of Audiologists|language=en-GB|access-date=2023-06-26}}</ref><ref>{{Cite web|url=https://www.ars.gov.hk/en/accr_pro_bodies.html|title=Accredited Registers Scheme for Healthcare Professions - Accredited Healthcare Professional Bodies|website=www.ars.gov.hk|access-date=2023-06-26}}</ref><ref>Press Releases, The Government of the Hong Kong SAR. Coverage of Elderly Health Care Voucher Scheme to extend to include four categories of healthcare profession under Accredited Registers Scheme for Healthcare Professions. 2023; Available at: <nowiki>https://www.info.gov.hk/gia/general/202304/27/P2023042700410.htm</nowiki>. Accessed Jun 21, 2023.</ref><ref>{{Cite web|url=https://www.deaf.org.hk/en/mission.php|title=The Hong Kong Society for the Deaf|website=www.deaf.org.hk|access-date=2023-06-26}}</ref><ref>{{Cite web|url=http://www.heartalk.org/en/about_us/our_mission/|title=Our Mission {{!}} Hear Talk Foundation|website=www.heartalk.org|access-date=2023-06-26}}</ref> <references /> {{Global Audiology Authors |name1=NG Hoi Yee Iris |role1=Author |website1=http://www.ihcr.cuhk.edu.hk/professor-iris-hoi-yee-ng/ |name2=Joyce Rodvie Sagun |role2=Contributor |linkedin2=http://linkedin.com/in/joyce-rodvie-sagun-4691bb182 }} tdsrtw6wxopvinwh65j0jszj0g5uq0i 0 317619 2806693 2806481 2026-04-26T19:13:37Z Redwidow448 3044853 Added my page for research 2806693 wikitext text/x-wiki Mathematics education is a discipline that studies the learning and teaching of mathematics.<ref>{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}</ref> == Mathematics Teacher Education Resources == * [[Levels of cognitive demand of tasks]]<ref>{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}</ref> * [[Lesson plans created by prospective teachers]] * [[Research: Technology's Support for Learning]] * [[Research: Manipulatives (Sydney)]] [[Category:Education]] [[Category:Mathematics|education]] <references /> * [[Research: Problem Based Learning]] darac6nxarnrowth1luu7g7yua69v4p 0 324670 2806740 2760183 2026-04-27T03:54:13Z OhanaUnited 18921 update 2806740 wikitext text/x-wiki <noinclude>{{WikiJHum top menu}} {{col-begin}} {{col-break}} </noinclude> {{Issue header|year = 2025 |volume = 6 |issue = 1 |current=false}} {{Article info |Q = Q136377790 |image=2025 Logo for the EduWiki Conference.png}} <noinclude>{{col-break}} {{WikiJHum right menu}} {{col-end}} [[category:{{ROOTPAGENAME}} issues]] </noinclude> 11bavs4p0bgk6jctexaa00icyxn2c7l 102 324936 2806780 2805281 2026-04-27T11:07:05Z Projet PEP 3002502 2806780 wikitext text/x-wiki <!-- BANNER ACROSS TOP OF PAGE --> <!---------------------- TABS -------------------------> {{Portal:Wikilang/start tab}} Welcome to the portal "Plurilingual education". It is a collection of free resources dedicated to plurilingual education to be used for pre-service and in-service training of language teachers. It has been created by the European project PEP, which is co-funded by the European Commission within the Erasmus+ programme (Promoting plurilingual education, 2023-1-FR01-KA220-HED-000160820). {{end tab}} <!---------------------- WELCOME -------------------------> {| cellspacing="0" cellpadding="0" | colspan="2" | |- style="vertical-align: top;" | style="padding-right: 1.2em; width: 50%;" | <!---------------------- FEATURED CONTENT -----------------------> {{Frame alt | color = 006699 | title = Featured resources | content = Following "lessons" are available. More are coming soon! * [[Awakening to languages]] * [[CLIL (Content and Language Integrated Learning)]] * [[Decolonial perspective in plurilingual education]] * [[Plurilingual education and digital technologies|Digital technologies in plurilingual education]] * [[Dominant language constellation]] * [[English as a Lingua Franca (ELF)]] * [[Endangered languages and plurilingual education]] * [[Heritage Language|Heritage language]] * [[Intercomprehension]] * [[Language biography and identity texts]] * [[Language inclusion]] * [[Language mediation]] * [[Language policies: Educational and family language policies]] * [[Language Portfolio|Language portfolio]] * [[Linguistic landscapes in education]] * [[Migrants, bilingualism & parental involvement]] * [[Multilingual awareness - Language awareness - Metacompetencies]] * [[Native speakerism]] * [[Non-formal and informal plurilingual education]] * [[Pluralistic approach]] * [[Pluringualism in the CEFR]] * [[Assessing the plurilingual competence|Plurilingual assessment - Assessing the plurilingual competence]] * [[Assessment of the knowledge and competences of plurilingual learners|Plurilingual assessment - Assessment of the knowledge and competences of plurilingual learners]] * [[Pedagogy of variation]] * [[Plurilingual and inter/transcultural competence]] * [[Plurilingualism and plurilingual education in the past]] * [[Telecollaboration and plurilingualism]] * [[Tertiary language teaching]] * [[Terminology and plurilingual education]] * [[Teachers’ beliefs and plurilingualism]] * [[Theories and models of plurilingualism]] * [[Translanguaging]] }} <!------------------------ LANGUAGES ---------------------------> {{Frame alt | color = 006699 | title = Library | content = Useful ressources to read * Cortés Velásquez, D., Strasser, M. et al. (2025a). ''L’utilisation des langues dans l’enseignement secondaire et supérieur : Croyances et pratiques des enseignants''. PEP – Promoting Plurilingual Education. [https://www.fdr.uni-hamburg.de/record/16757 https://www.fdr.uni-hamburg.de/record/16757] * Cortés Velásquez, D., Strasser, M. et al. (2025b). ''Project Promoting Plurilingual Education (PEP) -KA220-HED- E96C9232 Survey Report. Language use in secondary and higher education : Teachers’ beliefs and practices''. PEP – Promoting Plurilingual Education. [https://doi.org/10.25592/uhhfdm.16755 https://doi.org/10.25592/uhhfdm.16755] * Cortés Velásquez, D., Strasser, M. et al. (2025c). ''Sprachgebrauch in der Sekundar- und Hochschulbildung : Überzeugungen und Praktiken von Lehrkräften''. PEP – Promoting Plurilingual Education. [https://doi.org/10.25592/uhhfdm.16758 https://doi.org/10.25592/uhhfdm.16758 ] }} | style="padding-left: 1.2em; width: 50%;" | <!-------------------------- NEWS ---------------------------> {{Frame alt | color = 990000 | title = News | content = Selected worldwide news about plurilingual education: * '''[https://sites.google.com/view/pep-conference Conference - Bridging Voices in Plurilingual Education: Policies, Research and Practices]''', 23-24 october 2025, Rom. The conference was organised by Università degli Studi Roma Tre within the framework of the PEP project (Promoting plurilingual education, 2023-1-FR01-KA220-HED-000160820). }}<!------------------------ EXTERNAL RESOURCES ------------------------->{{Frame alt | title = External ressources | content = Projects and materials to "teach" plurilingual education *[https://sites.google.com/view/pepproject/productions/livret-de-bonnes-pratiques-good-practices-booklet Booklet of adaptable plurilingual practices] *[https://www.ecml.at/en/ECML-Programme/Programme-2020-2023/Mediation-in-teaching-and-assessment METLA - Mediation in teaching, learning and assessment] *[https://www.coe.int/en/web/language-policy/plurilingualism CEFR and Plurilingualism] *[https://carap.ecml.at/ CARAP/FREPA] }}<!------------------------ OTHER --------------------------->{{Frame alt | color = 339966 | title = Other resources in the Wikiversity | content = '''Learning Groups''' * [[Portal:Foreign Language Learning|Foreign Language Learning]] * [[Portal:TESOL|Teaching English to speakers of other languages (TESOL)]] * [[Portal:Translation|Translation]] '''In the French Wikiversité''' *[https://fr.wikiversity.org/wiki/D%C3%A9partement:Didactique_des_langues Department of plurilingual education in the French Wikiversité] }} [[Category:Wikilang|*]] [[Category:Foreign Language Learning|*]] [[fr:Faculté:Wikilangues]] <!-- {{Frame alt | color = 990000 | title = Click on a continent | content = <div>{{Wikilang map}}</div><br>Click on a continent to get to a portal of languages of this continent. }} |}--> |} g34fvqq0qoqr78d6qir0p51i25164nx 0 326035 2806743 2805016 2026-04-27T04:08:20Z RadiX 155307 minor fix 2806743 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Botswana (orthographic projection).svg|https://en.wikipedia.org/wiki/Botswana}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Botswana Botswana], officially the Republic of Botswana, is a landlocked country in Southern Africa. Botswana is topographically flat, with approximately 70% of its territory being a part of the Kalahari Desert. It is bordered by South Africa to the south and southeast, Namibia to the west and north, Zambia to the north, and Zimbabwe to the northeast. The official language of Botswana is English, while Setswana is widely spoken across the country. {{HTitle|History of Audiology}} {{HTitle|Incidence and Prevalence of Hearing Loss}} {{HTitle|Information About Audiology}} {{HTitle|Scope of Practice and Licensing}} {{HTitle|Professional and Regulatory Bodies}} {{HTitle|Ongoing audiology research}} {{HTitle|Challenges, Opportunities and Notes}} {{HTitle|Audiology Charities}} {{HTitle|References}} <references responsive="" /> {{:Global Audiology/Authors-3|ADD|NAMES|HERE|https://www.researchgate.net/profile/Stephanie-Borel-3|https://www.labo-audiologie-clinique.com/morganpotier|https://www.researchgate.net/profile/Hung-Thai-Van}} [[Category:Audiology]] [[Category:Botswana]] eqkpace956jcul6w5vrpa1x37dfyspy 2806765 2806743 2026-04-27T04:38:33Z RadiX 155307 upd 2806765 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Botswana (orthographic projection).svg|https://en.wikipedia.org/wiki/Botswana}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Botswana Botswana], officially the Republic of Botswana, is a landlocked country in Southern Africa. Botswana is topographically flat, with approximately 70% of its territory being a part of the Kalahari Desert. It is bordered by South Africa to the south and southeast, Namibia to the west and north, Zambia to the north, and Zimbabwe to the northeast. The official language of Botswana is English, while Setswana is widely spoken across the country. {{HTitle|History of Audiology}} {{HTitle|Incidence and Prevalence of Hearing Loss}} {{HTitle|Information About Audiology}} {{HTitle|Scope of Practice and Licensing}} {{HTitle|Professional and Regulatory Bodies}} {{HTitle|Ongoing audiology research}} {{HTitle|Challenges, Opportunities and Notes}} {{HTitle|Audiology Charities}} {{HTitle|References}} {{reflist}} {{Global Audiology Authors |name1=Stephanie Borel |role1=Author |researchgate1=https://www.researchgate.net/profile/Stephanie-Borel-3 |name2=Morgan Potier |role2=Author |website2=https://www.labo-audiologie-clinique.com/morganpotier |name3=Hung Thai-Van |role3=Author |researchgate3=https://www.researchgate.net/profile/Hung-Thai-Van }} [[Category:Audiology]] [[Category:Botswana]] 9fp7wzl8smfly9x3b32fa0f26kg146v 0 326316 2806758 2804978 2026-04-27T04:27:16Z RadiX 155307 +map 2806758 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Asia/Header}} {{CountryHeader|File:QAT orthographic.svg|https://en.wikipedia.org/wiki/Qatar}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Qatar Qatar], officially the State of Qatar, is a country in West Asia. It occupies the Qatar Peninsula on the northeastern coast of the Arabian Peninsula in the Middle East; it shares its sole land border with Saudi Arabia to the south, with the rest of its territory surrounded by the Persian Gulf. The Gulf of Bahrain, an inlet of the Persian Gulf, separates Qatar from nearby Bahrain. Arabic is the official language, with Qatari Arabic being the local dialect. English is widely used in business and education. Other languages such as Hindi, Urdu, Nepali, and Tagalog are common among its diverse expatriate communities. [https://en.wikipedia.org/wiki/Qatari_Unified_Sign_Language Qatari Sign Language] is the language of the native Qatari deaf community. {{HTitle|History of Audiology}}Audiology services in Qatar began in the early 2000s. Much of the progress was spearheaded by Dr Khalid Abdul Hadi, who established the Audiology and Balance Unit and laid the foundation for comprehensive hearing care. Under his leadership, [https://www.hamad.qa/EN/Pages/default.aspx Hamad Medical Corporation (HMC)] introduced advanced diagnostic and rehabilitative services, and launched Universal Newborn Hearing Screening in 2006, which became a national health policy, screening over 75,000 newborns for early detection of hearing loss by the year 2012 (Hadi et al., 2013). Dr Abdul Hadi also served as the National Lead for the Council of Persons with Disabilities, advocating for inclusive health policies and accessibility initiatives. His efforts positioned Qatar as a regional leader in hearing health and cochlear implant services, which began in 2004. Private sector contributions have also complemented public initiatives. Centers such as the [https://www.qish.info/ Qatar Institute for Speech and Hearing (QISH)] and [https://en.wikipedia.org/wiki/Al_Ahli_Hospital,_Hebron Al Ahli Hospital] are some of the earliest institutions in Qatar and have expanded from basic audiology and speech therapy to multidisciplinary rehabilitation services, including audiology, vestibular assessment, and auditory-verbal therapy. {{HTitle|Incidence and Prevalence of Hearing Loss}}Little is known about the actual epidemiology of hearing loss in Qatar, as comprehensive population-based studies are lacking. Existing data suggests a prevalence of approximately 5.2% in individuals born to parents of consanguineous marriages.<ref>{{Cite journal|last=Bener|first=Abdulbari|last2=Eihakeem|first2=Amr A. M.|last3=Abdulhadi|first3=Khaled|date=2005-03|title=Is there any association between consanguinity and hearing loss|url=https://pubmed.ncbi.nlm.nih.gov/15733591|journal=International Journal of Pediatric Otorhinolaryngology|volume=69|issue=3|pages=327–333|doi=10.1016/j.ijporl.2004.10.004|issn=0165-5876|pmid=15733591}}</ref> <ref>{{Cite journal|last=Girotto|first=Giorgia|last2=Mezzavilla|first2=Massimo|last3=Abdulhadi|first3=Khalid|last4=Vuckovic|first4=Dragana|last5=Vozzi|first5=Diego|last6=Khalifa Alkowari|first6=Moza|last7=Gasparini|first7=Paolo|last8=Badii|first8=Ramin|date=2014|title=Consanguinity and hereditary hearing loss in Qatar|url=https://pubmed.ncbi.nlm.nih.gov/25060281|journal=Human Heredity|volume=77|issue=1-4|pages=175–182|doi=10.1159/000360475|issn=1423-0062|pmid=25060281}}</ref> Genetic factors play a significant role due to high rates of consanguinity among the native population, which increases the risk of hereditary hearing loss (HHL).<ref>{{Cite journal|last=Alkowari|first=Moza K.|last2=Vozzi|first2=Diego|last3=Bhagat|first3=Shruti|last4=Krishnamoorthy|first4=Navaneethakrishnan|last5=Morgan|first5=Anna|last6=Hayder|first6=Yousra|last7=Logendra|first7=Barathy|last8=Najjar|first8=Nehal|last9=Gandin|first9=Ilaria|date=2017-08|title=Targeted sequencing identifies novel variants involved in autosomal recessive hereditary hearing loss in Qatari families|url=https://pubmed.ncbi.nlm.nih.gov/28501645|journal=Mutation Research|volume=800-802|pages=29–36|doi=10.1016/j.mrfmmm.2017.05.001|issn=1873-135X|pmid=28501645}}</ref> <ref>{{Cite journal|last=Alkhidir|first=Shaza|last2=El-Akouri|first2=Karen|last3=Al-Dewik|first3=Nader|last4=Khodjet-El-Khil|first4=Houssein|last5=Okashah|first5=Sarah|last6=Islam|first6=Nazmul|last7=Ben-Omran|first7=Tawfeg|last8=Al-Shafai|first8=Mashael|date=2024-02-20|title=The genetic basis and the diagnostic yield of genetic testing related to nonsyndromic hearing loss in Qatar|url=https://pubmed.ncbi.nlm.nih.gov/38378725|journal=Scientific Reports|volume=14|issue=1|pages=4202|doi=10.1038/s41598-024-52784-z|issn=2045-2322|pmc=10879212|pmid=38378725}}</ref> These findings highlight the need for robust epidemiological research and targeted genetic counseling to inform national strategies for prevention and early intervention of hearing loss. {{HTitle|Information About Audiology}}In both public and private sectors, audiologists in Qatar offer a comprehensive range of audiological services supported by advanced technology and skilled professionals. These services include: • Immittance Testing (tympanometry and acoustic reflexes) • Otoacoustic Emissions (OAE) for cochlear function assessment • Pure Tone Audiometry (air and bone conduction) • Speech Audiometry for speech perception and discrimination • Hearing Aid Fitting and Verification, including real-ear and test box measurements • Cochlear Implant Candidacy Assessment and post-implant programming (only offered at HMC Audiology & Balance Unit) • Auditory Evoked Potentials such as ABR (Auditory Brainstem Response), MLR (Middle Latency Response), and ALR (Auditory Late Response) • Vestibular Testing, including VEMP (Vestibular Evoked Myogenic Potentials) and VNG (Video nystagmography) {{HTitle|Scope of Practice and Licensing}} Audiology Technologists: Autonomous practice including prevention, diagnosis, and rehabilitation of auditory and vestibular disorders; newborn screening; hearing aid and cochlear implant management; vestibular rehabilitation; advocacy; and research. Audiology Technicians: Perform delegated tasks under supervision, such as otoscopy, pure-tone audiometry, tympanometry, and otoacoustic emissions testing. Both roles require licensure and continuing professional development [https://dhp.moph.gov.qa/en/Pages/HowToRegisterToPracticeInQatar.aspx Ministry of Public Health, Qatar]. '''Professionals providing hearing care services''' Hearing care in Qatar is delivered by a multidisciplinary team comprising: Audiovestibular Physicians: Medical specialists in auditory and vestibular disorders. * Audiology Technologists ([https://en.wikipedia.org/wiki/Audiology]): Licensed professionals providing diagnostic and rehabilitative services. * Audiology Technicians: Support staff performing delegated tasks under supervision. * Otolaryngologists, Otologists, and Otoneurologists: Physicians managing surgical interventions and complex vestibular disorders. * Speech-Language Pathologists: Provide auditory-verbal therapy for children with hearing loss. Services offered by Otolaryngologists, Otologists and Otoneurologists These specialists provide: • Surgical management of middle and inner ear disorders (myringotomy, tympanoplasty, ossiculoplasty, mastoidectomy, middle ear exploration, intratympanic steroid/antibiotic injections, tumour resections, eustachian tube dilation. • Implantation devices: Cochlear implant surgeries (HMC only), Osseo integrated implants (Sidra Medicine only), BAHA and postoperative care. • Diagnosis and treatment of vestibular disorders, including vestibular rehabilitation. According to the registered practitioners list from the Ministry of Public Health (January, 2026), the following numbers are currently registered: * Audiology Technologists:     50 * Audiology Technicians:       45 * Audio-vestibular physicians:  8 * Otolaryngologists:         245   '''Role of primary health care providers and community health workers in hearing care''' Audiologists employed at [https://www.phcc.gov.qa/health-centers Primary Healthcare Centres] (PHCC) facilitate early detection through screening programs and referrals to secondary and tertiary centres. As Doha and the Greater Doha areas are predominantly urban areas and have the highest population concentration, there are more than 31 PHCCs located across Qatar, with additional private health facilities providing both primary and secondary audiology services. The larger PHCCs, or Health Centres, typically have rotating Otolaryngology services to manage basic ENT conditions and will refer to higher levels of care when needed. Adults and adolescents are typically referred to the Audiology and Balance unit at the HMC Ambulatory Care Centre (Main medical hub- urban), Al Wakra Hospital (south-urban), Al Khor Hospital (north-rural), Aisha bint Hamad Al Attiyah Hospital (north peri-urban), or Cuban Hospital (east-rural). All these facilities are located across the country, allowing equitable access to care in both urban and rural areas. For paediatric and specialised cases, referrals are usually directed to Sidra Medicine, a non-profit for public benefit hospital, part of [https://en.wikipedia.org/wiki/Qatar_Foundation Qatar Foundation]. This referral network provides accessibility to free or highly subsidised specialist services even at a primary level for individuals of all ages. Laws related to hearing care services All healthcare professions, regardless of profession, are regulated by the Department of Healthcare Professions (DHP) under the [https://en.wikipedia.org/wiki/Ministry_of_Public_Health_(Qatar) Ministry of Public Health] (MOPH). Licensing, competency validation, and adherence to ethical and legal standards are mandatory for all practitioners. National health insurance ensures free access for Qatari citizens and subsidized care for expatriates ([https://hamad.qa/EN/Patient-Information/How-To-Get-A-Health-Card/Pages/default.aspx Hamad Medical Corporation, 2025]). '''Education and Professional Practice''' Education of professionals working in hearing care services Qatar currently lacks a dedicated audiology degree program. Most, if not all audiologists are recruited internationally, resulting in varied competencies. A three-tiered workforce structure exists: * Audio-vestibular Physicians- Medical practitioners specialized in audiological and vestibular disorders * Audiology Technologists – Bachelor’s or Master’s degree in audiology or related fields [https://dhp.moph.gov.qa/en/Pages/HowToRegisterToPracticeInQatar.aspx Ministry of Public Health, Qatar] (2020). * Audiology Technicians –Trained through a one-year diploma program. * Qatar University introduced a Bachelor’s Speech-Language Pathology program in 2023, signaling potential future expansion into audiology education ([https://www.qu.edu.qa/en-us/Colleges/chs/physical-therapy/Pages/speech-and-language-pathology.aspx Qatar University, 2025]). {{HTitle|Professional and Regulatory Bodies}}The Department of Healthcare Professions (DHP) regulates audiology practice, licensing, and competency standards. Professionals must meet educational and experiential criteria, pass qualifying examinations, and adhere to ethical and legal frameworks.{{HTitle|Ongoing audiology research}}Research in audiology within Qatar has primarily focused on consanguinity, genetics, and hereditary hearing loss, reflecting the high prevalence of genetic disorders in the population. Numerous studies have identified novel gene mutations associated with hearing impairment,<ref>{{Cite journal|last=Khalifa Alkowari|first=M.|last2=Girotto|first2=G.|last3=Abdulhadi|first3=K.|last4=Dipresa|first4=S.|last5=Siam|first5=R.|last6=Najjar|first6=N.|last7=Badii|first7=R.|last8=Gasparini|first8=P.|date=2012-03|title=GJB2 and GJB6 genes and the A1555G mitochondrial mutation are only minor causes of nonsyndromic hearing loss in the Qatari population|url=https://pubmed.ncbi.nlm.nih.gov/22103400|journal=International Journal of Audiology|volume=51|issue=3|pages=181–185|doi=10.3109/14992027.2011.625983|issn=1708-8186|pmid=22103400}}</ref> and new findings continue to emerge annually, highlighting Qatar’s contribution to global genetic research. In addition to genetic studies, newborn hearing screening has also been a research theme. Three key studies have evaluated newborn hearing screening programs over time: one from the public sector, <ref>{{Cite journal|last=Abdul Hadi|first=K.|last2=Salahaldin|first2=A.|last3=Al Qahtani|first3=A.|last4=Al Musleh|first4=Z.|last5=Al Sulaitin|first5=M.|last6=Bener|first6=A.|last7=Chandra|first7=P.|last8=Alawi|first8=F.|date=2012|title=Universal neonatal hearing screening: Six years of experience in Qatar|url=https://pubmed.ncbi.nlm.nih.gov/25003040|journal=Qatar Medical Journal|volume=2012|issue=2|pages=42–50|doi=10.5339/qmj.2012.2.12|issn=0253-8253|pmc=3991045|pmid=25003040}}</ref> another from the private sector,<ref>{{Cite journal|last=Elsanadiky|first=HanaaH|last2=Afifi|first2=PrettyO|date=2017|title=Universal neonatal hearing screening program in private hospital, Qatar|url=https://journals.lww.com/10.4103/tmj.tmj_24_17|journal=Tanta Medical Journal|language=en|volume=45|issue=4|pages=175|doi=10.4103/tmj.tmj_24_17|issn=1110-1415}}</ref> and a recent study from a semi-government hospital highlighting continuous progress in early detection and intervention.<ref>{{Cite journal|last=Andreas|first=Jarreth Noël|last2=Amde|first2=Woldekidan Kifle|last3=Roomaney|first3=Rifqah Abeeda|date=2025-12-15|title=Risk factors associated with hearing loss in neonates: A retrospective cross-sectional study from Qatar|url=https://www.qscience.com/content/journals/10.5339/qmj.2025.99|journal=Qatar Medical Journal|language=en|volume=2025|issue=4|doi=10.5339/qmj.2025.99|issn=0253-8253}}</ref> Other research areas include investigations into clinical associations with hearing loss, notably: * COVID-19 and auditory complications.<ref>{{Cite journal|last=Chandran|first=Reni K|last2=Abdulhadi|first2=Khalid|last3=Al-Shaikhly|first3=Sarah|last4=Arangodan|first4=Mohammed Ameen|last5=Ramadan|first5=Nadeen Mousa Issa|last6=Aldeeb|first6=Shahed Jehad Ahmad|last7=Sathian|first7=Brijesh|date=2024-12-24|title=Hearing Loss in COVID-19 Patients: An Audiological Profile of Symptomatic and Asymptomatic COVID-19 Patients in Qatar|url=https://www.cureus.com/articles/321104-hearing-loss-in-covid-19-patients-an-audiological-profile-of-symptomatic-and-asymptomatic-covid-19-patients-in-qatar|journal=Cureus|language=en|doi=10.7759/cureus.76326|issn=2168-8184|pmc=11756780|pmid=39850173}}</ref> * Ototoxicity related to pharmacological treatments.<ref>{{Cite journal|last=Al Musleh|first=Zainab|last2=Al Suliteen|first2=Maha|last3=Hadi|first3=Khalid|last4=El Abbadi|first4=Maysoun|last5=Omar|first5=Waleed|last6=Ali|first6=Awatif|last7=Al Amin|first7=Amna|last8=Alsufi|first8=Muna|last9=Alnajar|first9=Nehal|date=2016|title=Early detection of changes to hearing status attributed to treatment regimen with ototoxicity in the state of Qatar|url=http://www.aaj.eg.net/text.asp?2016/3/1/9/191236|journal=Advanced Arab Academy of Audio-Vestibulogy Journal|language=en|volume=3|issue=1|pages=9|doi=10.4103/2314-8667.191236|issn=2314-8667}}</ref> * Presbycusis, balance and age-related hearing decline.<ref>{{Cite journal|last=Omer|first=Walid E.|last2=Abdulhadi|first2=Khalid|last3=Shahbal|first3=Saad|last4=Neudert|first4=Marcus|last5=Siepmann|first5=Timo|date=2024-12-19|title=Vestibular Hypofunction in Patients with Presbycusis: A Cross-Sectional Study|url=https://pubmed.ncbi.nlm.nih.gov/39768690|journal=Journal of Clinical Medicine|volume=13|issue=24|pages=7767|doi=10.3390/jcm13247767|issn=2077-0383|pmc=11727745|pmid=39768690}}</ref> Clinician-led research by audiologists are however lacking and greatly needed. {{HTitle|Challenges, Opportunities and Notes}}Healthcare in Qatar is well-funded and all residents have good access regardless of social standing. Due to the large number of expatriate workers, especially considering the country of origin, there may be differences in standardization of practice, particularly where protocols and guidelines relating to audiology and hearing care is affected. The Department of Healthcare Professions (DHP) regulates all healthcare professions in Qatar, however, Qatar may benefit from a local Audiological society which may help assist governmental organizations in making decisions regarding the profession. It could also assist in networking for case discussions, specialized continuous professional development, and putting a name to the face when referring across institutions. Clinician-led research appears to be lacking. This will aid in understanding potential challenges and opportunities for growth on a grassroots level. Many audiologists working in Qatar have postgraduate training (e.g. Masters), which includes have research experience. Further championing in this regard will help position the Audiology profession well in the country. Lastly, the establishment of an Audiology programme at one of the local universities will further aid in ensuring the profession continues to thrive, is backed by research, and ensures that the next generation has an opportunity to pursue Audiology as a profession. The recruitment and onboarding of audiologists from abroad can be costly to employers, thus with the establishment of a local audiology programme, this ensures that local organizations have a steady supply of healthcare professionals, which will not only be cost-effective to the local economy, but also assist exporting countries in reducing brain-drain and transfer of skills abroad. {{HTitle|Audiology Charities}}There are no specific audiological charities in Qatar, however, several charitable organizations play a vital role in supporting individuals who cannot afford audiological services or assistive devices. The four main charities include: * [https://en.wikipedia.org/wiki/Qatar_Charity Qatar Charity] * [https://en.wikipedia.org/wiki/Qatar_Red_Crescent_Society Qatar Red Crescent Society] * Qatar Society for the Rehabilitation of Special Needs * Qatar Cancer Society These organizations are typically funded through a combination of government allocations, private benefactors, and Zakat contributions. When patients face financial barriers to accessing hearing care or associated treatments, they often approach these charities for assistance. Support is frequently coordinated through hospital social workers, who facilitate applications and liaise with charitable bodies to ensure timely provision of services and devices. {{HTitle|References}} {{reflist}} {{:Global Audiology/Authors-1|Jarreth N. Andreas|https://www.researchgate.net/profile/Jarreth-Andreas}} ''Edited by'' [https://orcid.org/0000-0002-4586-2398/ Nausheen Dawood] [[Category:Audiology]] [[Category:Qatar]] hd6ai9nzkasb7x9klhy5p8zecr7f1lw 2806760 2806758 2026-04-27T04:29:20Z RadiX 155307 upd 2806760 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Asia/Header}} {{CountryHeader|File:QAT orthographic.svg|https://en.wikipedia.org/wiki/Qatar}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Qatar Qatar], officially the State of Qatar, is a country in West Asia. It occupies the Qatar Peninsula on the northeastern coast of the Arabian Peninsula in the Middle East; it shares its sole land border with Saudi Arabia to the south, with the rest of its territory surrounded by the Persian Gulf. The Gulf of Bahrain, an inlet of the Persian Gulf, separates Qatar from nearby Bahrain. Arabic is the official language, with Qatari Arabic being the local dialect. English is widely used in business and education. Other languages such as Hindi, Urdu, Nepali, and Tagalog are common among its diverse expatriate communities. [https://en.wikipedia.org/wiki/Qatari_Unified_Sign_Language Qatari Sign Language] is the language of the native Qatari deaf community. {{HTitle|History of Audiology}}Audiology services in Qatar began in the early 2000s. Much of the progress was spearheaded by Dr Khalid Abdul Hadi, who established the Audiology and Balance Unit and laid the foundation for comprehensive hearing care. Under his leadership, [https://www.hamad.qa/EN/Pages/default.aspx Hamad Medical Corporation (HMC)] introduced advanced diagnostic and rehabilitative services, and launched Universal Newborn Hearing Screening in 2006, which became a national health policy, screening over 75,000 newborns for early detection of hearing loss by the year 2012 (Hadi et al., 2013). Dr Abdul Hadi also served as the National Lead for the Council of Persons with Disabilities, advocating for inclusive health policies and accessibility initiatives. His efforts positioned Qatar as a regional leader in hearing health and cochlear implant services, which began in 2004. Private sector contributions have also complemented public initiatives. Centers such as the [https://www.qish.info/ Qatar Institute for Speech and Hearing (QISH)] and [https://en.wikipedia.org/wiki/Al_Ahli_Hospital,_Hebron Al Ahli Hospital] are some of the earliest institutions in Qatar and have expanded from basic audiology and speech therapy to multidisciplinary rehabilitation services, including audiology, vestibular assessment, and auditory-verbal therapy. {{HTitle|Incidence and Prevalence of Hearing Loss}}Little is known about the actual epidemiology of hearing loss in Qatar, as comprehensive population-based studies are lacking. Existing data suggests a prevalence of approximately 5.2% in individuals born to parents of consanguineous marriages.<ref>{{Cite journal|last=Bener|first=Abdulbari|last2=Eihakeem|first2=Amr A. M.|last3=Abdulhadi|first3=Khaled|date=2005-03|title=Is there any association between consanguinity and hearing loss|url=https://pubmed.ncbi.nlm.nih.gov/15733591|journal=International Journal of Pediatric Otorhinolaryngology|volume=69|issue=3|pages=327–333|doi=10.1016/j.ijporl.2004.10.004|issn=0165-5876|pmid=15733591}}</ref> <ref>{{Cite journal|last=Girotto|first=Giorgia|last2=Mezzavilla|first2=Massimo|last3=Abdulhadi|first3=Khalid|last4=Vuckovic|first4=Dragana|last5=Vozzi|first5=Diego|last6=Khalifa Alkowari|first6=Moza|last7=Gasparini|first7=Paolo|last8=Badii|first8=Ramin|date=2014|title=Consanguinity and hereditary hearing loss in Qatar|url=https://pubmed.ncbi.nlm.nih.gov/25060281|journal=Human Heredity|volume=77|issue=1-4|pages=175–182|doi=10.1159/000360475|issn=1423-0062|pmid=25060281}}</ref> Genetic factors play a significant role due to high rates of consanguinity among the native population, which increases the risk of hereditary hearing loss (HHL).<ref>{{Cite journal|last=Alkowari|first=Moza K.|last2=Vozzi|first2=Diego|last3=Bhagat|first3=Shruti|last4=Krishnamoorthy|first4=Navaneethakrishnan|last5=Morgan|first5=Anna|last6=Hayder|first6=Yousra|last7=Logendra|first7=Barathy|last8=Najjar|first8=Nehal|last9=Gandin|first9=Ilaria|date=2017-08|title=Targeted sequencing identifies novel variants involved in autosomal recessive hereditary hearing loss in Qatari families|url=https://pubmed.ncbi.nlm.nih.gov/28501645|journal=Mutation Research|volume=800-802|pages=29–36|doi=10.1016/j.mrfmmm.2017.05.001|issn=1873-135X|pmid=28501645}}</ref> <ref>{{Cite journal|last=Alkhidir|first=Shaza|last2=El-Akouri|first2=Karen|last3=Al-Dewik|first3=Nader|last4=Khodjet-El-Khil|first4=Houssein|last5=Okashah|first5=Sarah|last6=Islam|first6=Nazmul|last7=Ben-Omran|first7=Tawfeg|last8=Al-Shafai|first8=Mashael|date=2024-02-20|title=The genetic basis and the diagnostic yield of genetic testing related to nonsyndromic hearing loss in Qatar|url=https://pubmed.ncbi.nlm.nih.gov/38378725|journal=Scientific Reports|volume=14|issue=1|pages=4202|doi=10.1038/s41598-024-52784-z|issn=2045-2322|pmc=10879212|pmid=38378725}}</ref> These findings highlight the need for robust epidemiological research and targeted genetic counseling to inform national strategies for prevention and early intervention of hearing loss. {{HTitle|Information About Audiology}}In both public and private sectors, audiologists in Qatar offer a comprehensive range of audiological services supported by advanced technology and skilled professionals. These services include: • Immittance Testing (tympanometry and acoustic reflexes) • Otoacoustic Emissions (OAE) for cochlear function assessment • Pure Tone Audiometry (air and bone conduction) • Speech Audiometry for speech perception and discrimination • Hearing Aid Fitting and Verification, including real-ear and test box measurements • Cochlear Implant Candidacy Assessment and post-implant programming (only offered at HMC Audiology & Balance Unit) • Auditory Evoked Potentials such as ABR (Auditory Brainstem Response), MLR (Middle Latency Response), and ALR (Auditory Late Response) • Vestibular Testing, including VEMP (Vestibular Evoked Myogenic Potentials) and VNG (Video nystagmography) {{HTitle|Scope of Practice and Licensing}} Audiology Technologists: Autonomous practice including prevention, diagnosis, and rehabilitation of auditory and vestibular disorders; newborn screening; hearing aid and cochlear implant management; vestibular rehabilitation; advocacy; and research. Audiology Technicians: Perform delegated tasks under supervision, such as otoscopy, pure-tone audiometry, tympanometry, and otoacoustic emissions testing. Both roles require licensure and continuing professional development [https://dhp.moph.gov.qa/en/Pages/HowToRegisterToPracticeInQatar.aspx Ministry of Public Health, Qatar]. '''Professionals providing hearing care services''' Hearing care in Qatar is delivered by a multidisciplinary team comprising: Audiovestibular Physicians: Medical specialists in auditory and vestibular disorders. * Audiology Technologists ([https://en.wikipedia.org/wiki/Audiology]): Licensed professionals providing diagnostic and rehabilitative services. * Audiology Technicians: Support staff performing delegated tasks under supervision. * Otolaryngologists, Otologists, and Otoneurologists: Physicians managing surgical interventions and complex vestibular disorders. * Speech-Language Pathologists: Provide auditory-verbal therapy for children with hearing loss. Services offered by Otolaryngologists, Otologists and Otoneurologists These specialists provide: • Surgical management of middle and inner ear disorders (myringotomy, tympanoplasty, ossiculoplasty, mastoidectomy, middle ear exploration, intratympanic steroid/antibiotic injections, tumour resections, eustachian tube dilation. • Implantation devices: Cochlear implant surgeries (HMC only), Osseo integrated implants (Sidra Medicine only), BAHA and postoperative care. • Diagnosis and treatment of vestibular disorders, including vestibular rehabilitation. According to the registered practitioners list from the Ministry of Public Health (January, 2026), the following numbers are currently registered: * Audiology Technologists:     50 * Audiology Technicians:       45 * Audio-vestibular physicians:  8 * Otolaryngologists:         245   '''Role of primary health care providers and community health workers in hearing care''' Audiologists employed at [https://www.phcc.gov.qa/health-centers Primary Healthcare Centres] (PHCC) facilitate early detection through screening programs and referrals to secondary and tertiary centres. As Doha and the Greater Doha areas are predominantly urban areas and have the highest population concentration, there are more than 31 PHCCs located across Qatar, with additional private health facilities providing both primary and secondary audiology services. The larger PHCCs, or Health Centres, typically have rotating Otolaryngology services to manage basic ENT conditions and will refer to higher levels of care when needed. Adults and adolescents are typically referred to the Audiology and Balance unit at the HMC Ambulatory Care Centre (Main medical hub- urban), Al Wakra Hospital (south-urban), Al Khor Hospital (north-rural), Aisha bint Hamad Al Attiyah Hospital (north peri-urban), or Cuban Hospital (east-rural). All these facilities are located across the country, allowing equitable access to care in both urban and rural areas. For paediatric and specialised cases, referrals are usually directed to Sidra Medicine, a non-profit for public benefit hospital, part of [https://en.wikipedia.org/wiki/Qatar_Foundation Qatar Foundation]. This referral network provides accessibility to free or highly subsidised specialist services even at a primary level for individuals of all ages. Laws related to hearing care services All healthcare professions, regardless of profession, are regulated by the Department of Healthcare Professions (DHP) under the [https://en.wikipedia.org/wiki/Ministry_of_Public_Health_(Qatar) Ministry of Public Health] (MOPH). Licensing, competency validation, and adherence to ethical and legal standards are mandatory for all practitioners. National health insurance ensures free access for Qatari citizens and subsidized care for expatriates ([https://hamad.qa/EN/Patient-Information/How-To-Get-A-Health-Card/Pages/default.aspx Hamad Medical Corporation, 2025]). '''Education and Professional Practice''' Education of professionals working in hearing care services Qatar currently lacks a dedicated audiology degree program. Most, if not all audiologists are recruited internationally, resulting in varied competencies. A three-tiered workforce structure exists: * Audio-vestibular Physicians- Medical practitioners specialized in audiological and vestibular disorders * Audiology Technologists – Bachelor’s or Master’s degree in audiology or related fields [https://dhp.moph.gov.qa/en/Pages/HowToRegisterToPracticeInQatar.aspx Ministry of Public Health, Qatar] (2020). * Audiology Technicians –Trained through a one-year diploma program. * Qatar University introduced a Bachelor’s Speech-Language Pathology program in 2023, signaling potential future expansion into audiology education ([https://www.qu.edu.qa/en-us/Colleges/chs/physical-therapy/Pages/speech-and-language-pathology.aspx Qatar University, 2025]). {{HTitle|Professional and Regulatory Bodies}}The Department of Healthcare Professions (DHP) regulates audiology practice, licensing, and competency standards. Professionals must meet educational and experiential criteria, pass qualifying examinations, and adhere to ethical and legal frameworks.{{HTitle|Ongoing audiology research}}Research in audiology within Qatar has primarily focused on consanguinity, genetics, and hereditary hearing loss, reflecting the high prevalence of genetic disorders in the population. Numerous studies have identified novel gene mutations associated with hearing impairment,<ref>{{Cite journal|last=Khalifa Alkowari|first=M.|last2=Girotto|first2=G.|last3=Abdulhadi|first3=K.|last4=Dipresa|first4=S.|last5=Siam|first5=R.|last6=Najjar|first6=N.|last7=Badii|first7=R.|last8=Gasparini|first8=P.|date=2012-03|title=GJB2 and GJB6 genes and the A1555G mitochondrial mutation are only minor causes of nonsyndromic hearing loss in the Qatari population|url=https://pubmed.ncbi.nlm.nih.gov/22103400|journal=International Journal of Audiology|volume=51|issue=3|pages=181–185|doi=10.3109/14992027.2011.625983|issn=1708-8186|pmid=22103400}}</ref> and new findings continue to emerge annually, highlighting Qatar’s contribution to global genetic research. In addition to genetic studies, newborn hearing screening has also been a research theme. Three key studies have evaluated newborn hearing screening programs over time: one from the public sector, <ref>{{Cite journal|last=Abdul Hadi|first=K.|last2=Salahaldin|first2=A.|last3=Al Qahtani|first3=A.|last4=Al Musleh|first4=Z.|last5=Al Sulaitin|first5=M.|last6=Bener|first6=A.|last7=Chandra|first7=P.|last8=Alawi|first8=F.|date=2012|title=Universal neonatal hearing screening: Six years of experience in Qatar|url=https://pubmed.ncbi.nlm.nih.gov/25003040|journal=Qatar Medical Journal|volume=2012|issue=2|pages=42–50|doi=10.5339/qmj.2012.2.12|issn=0253-8253|pmc=3991045|pmid=25003040}}</ref> another from the private sector,<ref>{{Cite journal|last=Elsanadiky|first=HanaaH|last2=Afifi|first2=PrettyO|date=2017|title=Universal neonatal hearing screening program in private hospital, Qatar|url=https://journals.lww.com/10.4103/tmj.tmj_24_17|journal=Tanta Medical Journal|language=en|volume=45|issue=4|pages=175|doi=10.4103/tmj.tmj_24_17|issn=1110-1415}}</ref> and a recent study from a semi-government hospital highlighting continuous progress in early detection and intervention.<ref>{{Cite journal|last=Andreas|first=Jarreth Noël|last2=Amde|first2=Woldekidan Kifle|last3=Roomaney|first3=Rifqah Abeeda|date=2025-12-15|title=Risk factors associated with hearing loss in neonates: A retrospective cross-sectional study from Qatar|url=https://www.qscience.com/content/journals/10.5339/qmj.2025.99|journal=Qatar Medical Journal|language=en|volume=2025|issue=4|doi=10.5339/qmj.2025.99|issn=0253-8253}}</ref> Other research areas include investigations into clinical associations with hearing loss, notably: * COVID-19 and auditory complications.<ref>{{Cite journal|last=Chandran|first=Reni K|last2=Abdulhadi|first2=Khalid|last3=Al-Shaikhly|first3=Sarah|last4=Arangodan|first4=Mohammed Ameen|last5=Ramadan|first5=Nadeen Mousa Issa|last6=Aldeeb|first6=Shahed Jehad Ahmad|last7=Sathian|first7=Brijesh|date=2024-12-24|title=Hearing Loss in COVID-19 Patients: An Audiological Profile of Symptomatic and Asymptomatic COVID-19 Patients in Qatar|url=https://www.cureus.com/articles/321104-hearing-loss-in-covid-19-patients-an-audiological-profile-of-symptomatic-and-asymptomatic-covid-19-patients-in-qatar|journal=Cureus|language=en|doi=10.7759/cureus.76326|issn=2168-8184|pmc=11756780|pmid=39850173}}</ref> * Ototoxicity related to pharmacological treatments.<ref>{{Cite journal|last=Al Musleh|first=Zainab|last2=Al Suliteen|first2=Maha|last3=Hadi|first3=Khalid|last4=El Abbadi|first4=Maysoun|last5=Omar|first5=Waleed|last6=Ali|first6=Awatif|last7=Al Amin|first7=Amna|last8=Alsufi|first8=Muna|last9=Alnajar|first9=Nehal|date=2016|title=Early detection of changes to hearing status attributed to treatment regimen with ototoxicity in the state of Qatar|url=http://www.aaj.eg.net/text.asp?2016/3/1/9/191236|journal=Advanced Arab Academy of Audio-Vestibulogy Journal|language=en|volume=3|issue=1|pages=9|doi=10.4103/2314-8667.191236|issn=2314-8667}}</ref> * Presbycusis, balance and age-related hearing decline.<ref>{{Cite journal|last=Omer|first=Walid E.|last2=Abdulhadi|first2=Khalid|last3=Shahbal|first3=Saad|last4=Neudert|first4=Marcus|last5=Siepmann|first5=Timo|date=2024-12-19|title=Vestibular Hypofunction in Patients with Presbycusis: A Cross-Sectional Study|url=https://pubmed.ncbi.nlm.nih.gov/39768690|journal=Journal of Clinical Medicine|volume=13|issue=24|pages=7767|doi=10.3390/jcm13247767|issn=2077-0383|pmc=11727745|pmid=39768690}}</ref> Clinician-led research by audiologists are however lacking and greatly needed. {{HTitle|Challenges, Opportunities and Notes}}Healthcare in Qatar is well-funded and all residents have good access regardless of social standing. Due to the large number of expatriate workers, especially considering the country of origin, there may be differences in standardization of practice, particularly where protocols and guidelines relating to audiology and hearing care is affected. The Department of Healthcare Professions (DHP) regulates all healthcare professions in Qatar, however, Qatar may benefit from a local Audiological society which may help assist governmental organizations in making decisions regarding the profession. It could also assist in networking for case discussions, specialized continuous professional development, and putting a name to the face when referring across institutions. Clinician-led research appears to be lacking. This will aid in understanding potential challenges and opportunities for growth on a grassroots level. Many audiologists working in Qatar have postgraduate training (e.g. Masters), which includes have research experience. Further championing in this regard will help position the Audiology profession well in the country. Lastly, the establishment of an Audiology programme at one of the local universities will further aid in ensuring the profession continues to thrive, is backed by research, and ensures that the next generation has an opportunity to pursue Audiology as a profession. The recruitment and onboarding of audiologists from abroad can be costly to employers, thus with the establishment of a local audiology programme, this ensures that local organizations have a steady supply of healthcare professionals, which will not only be cost-effective to the local economy, but also assist exporting countries in reducing brain-drain and transfer of skills abroad. {{HTitle|Audiology Charities}}There are no specific audiological charities in Qatar, however, several charitable organizations play a vital role in supporting individuals who cannot afford audiological services or assistive devices. The four main charities include: * [https://en.wikipedia.org/wiki/Qatar_Charity Qatar Charity] * [https://en.wikipedia.org/wiki/Qatar_Red_Crescent_Society Qatar Red Crescent Society] * Qatar Society for the Rehabilitation of Special Needs * Qatar Cancer Society These organizations are typically funded through a combination of government allocations, private benefactors, and Zakat contributions. When patients face financial barriers to accessing hearing care or associated treatments, they often approach these charities for assistance. Support is frequently coordinated through hospital social workers, who facilitate applications and liaise with charitable bodies to ensure timely provision of services and devices. {{HTitle|References}} {{reflist}} {{Global Audiology Authors |name1=Jarreth N. Andreas |role1=Author |researchgate1=https://www.researchgate.net/profile/Jarreth-Andreas |name2=Nausheen Dawood |role2=Contributor |orcid2=https://orcid.org/0000-0002-4586-2398/ }} [[Category:Audiology]] [[Category:Qatar]] 8uv42p6vpezw39f2c4343xl6ybgxoq7 0 326317 2806752 2804983 2026-04-27T04:19:27Z RadiX 155307 fix 2806752 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Asia/Header}} {{CountryHeader|File:State of Palestine (orthographic projection).svg|https://en.wikipedia.org/wiki/Palestine}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Palestine Palestine], officially the State of [[Palestine]], is a partially recognized sovereign state in the Levant region of Western Asia. It encompasses the Israeli-occupied West Bank, including East Jerusalem, and the Gaza Strip, collectively known as the Palestinian territories. The territories share the vast majority of their borders with Israel, with the West Bank bordering Jordan to the east and the Gaza Strip bordering Egypt to the southwest. [[Arabic]] is the official language of the State of Palestine, specifically the Palestinian Arabic dialect. Hebrew and English are also widely spoken. {{HTitle|History of Audiology}} Audiology is a relatively new health profession in the West Bank and Gaza. The field has expanded gradually over the past three decades through university programs and the establishment of local clinics. Some hearing care services have also been supported through collaborations with international organizations, enabling the provision of hearing aids, cochlear implants, and otologic surgeries. Despite these developments, many towns and villages still lack dedicated audiology clinics, leaving large segments of the population underserved. {{HTitle|Incidence and Prevalence of Hearing Loss}} To date, no national epidemiological survey conducted by the [https://en.wikipedia.org/wiki/Ministry_of_Health_(Palestine)|Palestinian Ministry of Health] has comprehensively measured the incidence or prevalence of hearing loss across the West Bank and Gaza Strip ([https://moh.ps/mohsite/Content/Books/gaWxiN5DMkd4v4LhRW8YB5BqufSinMh9gJKgHIo6sS56PgPHOFb4ri%20mQCSoDDSxCysOckjMPMFbBJfuqS6dNU86jCVIpfehsjOVkQZrqDqZp.pdf Annual Health Report, 2024]). However, the only sustained newborn hearing screening initiative to date was established by [https://en.wikipedia.org/wiki/Caritas_Baby_Hospital Caritas Baby Hospital] in Bethlehem. Between September 25, 2006, and December 31, 2011, Caritas screened more than 15,000 newborns, reporting a higher-than-expected prevalence of infant hearing loss compared with global estimates, highlighting the urgent need for a national screening system.<ref>{{Cite journal|last=Corradin|first=Lucia|last2=Hindiyeh|first2=Musa|last3=Khaled|first3=Rasha|last4=Rishmawi|first4=Fadi|last5=Zidan|first5=Marwan|last6=Marzouqa|first6=Hiyam|date=2014-11-21|title=Survey on Infant Hearing Loss at Caritas Baby Hospital in Bethlehem-Palestine|url=https://www.mdpi.com/2039-4349/4/1/99|journal=Audiology Research|language=en|volume=4|issue=1|pages=99|doi=10.4081/audiores.2014.99|issn=2039-4349|pmc=4627132|pmid=26557353}}</ref> {{HTitle|Information About Audiology}} Hearing Care Services Professionals Audiology and hearing services are provided by audiologists, speech-language therapists, and ENT (otolaryngology) physicians across governmental, NGO, and private settings. Audiological Services * Diagnostic services: puretone audiometry, tympanometry, acoustic reflex testing, otoacoustic emissions (OAE), and auditory brainstem response (ABR). * Hearing aid services: fitting, programming, maintenance, repair, and earmold production. * Cochlear implant services: mapping, troubleshooting, and device maintenance. * Hearing rehabilitation: auditory training and speech-language therapy. * Vestibular assessment: limited availability of videonystagmography (VNG) in a few private clinics. ENT / Otologists / Otoneurologists ENT surgeons provide a range of surgical and medical services, including cochlear implantations and middle-ear surgeries. Some CI surgeries are conducted through collaborative or mission-based programs.<ref>{{Cite web|url=https://reliefweb.int/report/occupied-palestinian-territory/qrcs-provides-hearing-aids-gaza-newborn-children-enar|title=QRCS provides hearing aids for Gaza newborn children [EN/AR] - occupied Palestinian territory {{!}} ReliefWeb|date=2022-09-21|website=reliefweb.int|language=en|access-date=2025-12-13}}</ref> <ref>{{Cite web|url=https://alameen.ngo/announcing-the-opening-of-registration-for-the-auditory-rehabilitation-and-cochlear-implant-program-for-palestinian-children-residing-in-the-middle-east/|title=Announcing the opening of registration for the Auditory Rehabilitation and Cochlear Implant Program for Palestinian children residing in the Middle East|last=Nadeem|date=2024-06-26|website=IRVD|language=en-US|access-date=2025-12-13}}</ref> <ref>{{Cite web|url=https://spa.gov.sa/|title=KSrelief Volunteer Medical Team Carries out 40 Cochlear Implants for Palestinian Children|website=spa.gov.sa|language=en|access-date=2025-12-13}}</ref> <ref>{{Cite web|url=https://reliefweb.int/report/occupied-palestinian-territory/qrcs-provides-hearing-aids-gaza-newborn-children-enar|title=QRCS provides hearing aids for Gaza newborn children [EN/AR] - occupied Palestinian territory {{!}} ReliefWeb|date=2022-09-21|website=reliefweb.int|language=en|access-date=2025-12-13}}</ref> Services—such as balance and tinnitus evaluation—exist but remain limited and unevenly distributed. Primary Health Care Primary care centers refer individuals suspected to have hearing impairment; however, universal newborn hearing screening (UNHS) is still not implemented nationwide. There is no dedicated national law regulating audiology practice. Clinicians providing hearing care must hold a valid license, and all clinics require Ministry of Health approval. {{HTitle|Scope of Practice and Licensing}} Education and Professional Practice Most audiologists hold a Bachelor’s degree in Speech Therapy and Audiology, with some holding diplomas from institutions accredited by the Palestinian Ministry of Higher Education, including [https://en.wikipedia.org/wiki/An-Najah_National_University An-Najah National University], [https://en.wikipedia.org/wiki/Birzeit_University|Birzeit Birzeit University], [https://en.wikipedia.org/wiki/Bethlehem_University|Bethlehem Bethlehem University], [https://en.wikipedia.org/wiki/Arab%20American%20University%20(Palestine) Arab American University], [https://en.wikipedia.org/wiki/Al-Quds_University|Al-Quds Al-Quds University], and Al-Terra College. However, there are no fully specialized or advanced audiology degree programs (Master’s or PhD) currently offered. Professional and regulatory bodies Audiology and speech-language therapy are not yet overseen by a unified national regulatory body. Licensing is processed through the Palestinian Ministry of Health for graduates of approved programs. A new requirement introduced in October 2025 obligates newly graduated or previously unlicensed practitioners to pass a licensing examination. Audiologists’ scope of practice typically includes diagnostics, hearing aid services, and rehabilitation. Surgical interventions remain the domain of ENT specialists. {{HTitle|Professional and Regulatory Bodies}} There is no dedicated national research society for audiology. Research efforts are fragmented and largely driven by universities, hospitals, or individual graduate students. Only a small number of studies related to hearing and audiology have been conducted, and many have not been formally published. {{HTitle|Research in Audiology}} There is no dedicated national research society for audiology. Research efforts are fragmented and largely driven by universities, hospitals, or individual graduate students. Only a small number of studies related to hearing and audiology have been conducted, and many have not been formally published.<ref>{{Cite journal|last=El-Dabbakeh|first=Hatem S|date=2024-10-14|title=Risk Factors for Hearing Impairment Among Primary School Deaf Children in Gaza Strip, Palestine|url=https://journal.medtigo.com/risk-factors-for-hearing-impairment-among-primary-school-deaf-children-in-gaza-strip-palestine/|journal=medtigo Journal of Medicine|language=en|volume=1|issue=1|pages=1–5|doi=10.63096/medtigo30622423|issn=3066-3202}}</ref> <ref>{{Cite journal|last=Abu Rayyan|first=Amal|last2=Kamal|first2=Lara|last3=Casadei|first3=Silvia|last4=Brownstein|first4=Zippora|last5=Zahdeh|first5=Fouad|last6=Shahin|first6=Hashem|last7=Canavati|first7=Christina|last8=Dweik|first8=Dima|last9=Jaraysa|first9=Tamara|date=2020-08-18|title=Genomic analysis of inherited hearing loss in the Palestinian population|url=https://pnas.org/doi/full/10.1073/pnas.2009628117|journal=Proceedings of the National Academy of Sciences|language=en|volume=117|issue=33|pages=20070–20076|doi=10.1073/pnas.2009628117|issn=0027-8424|pmc=7443947|pmid=32747562}}</ref> {{HTitle|Challenges, Opportunities and Notes}} '''Challenges''' * Uneven geographic distribution of specialized services, with many remote areas underserved. * Heavy reliance on external funding for hearing aids and cochlear implants. * Absence of universal newborn hearing screening and standardized early detection protocols. * High genetic burden of hearing loss due to consanguinity, increasing the need for genetic counseling. * Growing number of audiology graduates versus a limited number of clinics. * Limited validated data and research on hearing health. * Lack of public awareness and early intervention advocacy campaigns. * Insufficient clinical training opportunities and hands-on practice for students and new graduates. * Weak regulatory oversight of clinics and professionals; some centers employ unqualified people. * Movement restrictions and import limitations affecting the access of hearing equipment and care. * Scarcity of specialists with advanced degrees in audiology. * Financial barriers for families seeking hearing aids or cochlear implants. * Social stigma surrounding hearing devices, contributing to delayed intervention and detection. '''Opportunities''' * New academic programs in audiology at Palestinian universities open the way for more local professionals * Collaboration with NGOs and international partners to implement newborn hearing screening and protocols * Development of a national regulatory body to standardize practice and licensing. * Strengthening Palestine Ministry of Health oversight and accountability. * Establishing additional hearing care clinics through local or international partnerships. {{HTitle|Impact of the war on hearing loss}} Prolonged armed conflict, repeated military incursions in the West Bank, and the ongoing war in the Gaza Strip have had profound and cumulative effects on hearing health, particularly among persons with hearing disabilities. Continuous exposure to high intensity noise sources—such as airstrikes, artillery fire, sound bombs, live ammunition, and explosions—has been linked to acoustic trauma, tinnitus, and temporary or permanent hearing loss among both children and adults [UN News, 2024]. In Gaza, the low flying military aircraft, drones, and surveillance planes that operate continuously, sometimes for 24 hours a day, have been argued to contribute to persistent tinnitus, auditory fatigue, sleep disturbances, and stress related auditory symptoms. Recent local reports estimated that as many as 35,000 individuals in Gaza are experience hearing problems attributable to the extreme exposure from bombings, underscoring the severity of war-related auditory impacts on civilians. Reports from international and local organizations indicate that the documented prevalence of hearing loss in Gaza and the West Bank significantly underestimates the true burden, as access to audiological assessment, diagnosis, and rehabilitation services is extremely limited or entirely unavailable during periods of active conflict.<ref>{{Cite journal|last=Pakulski|first=Lori A.|date=2024-08|title=Addressing Hearing Loss of Palestinians Living in Refugee Camps|url=https://pubs.asha.org/doi/10.1044/2024_PERSP-23-00251|journal=Perspectives of the ASHA Special Interest Groups|language=en|volume=9|issue=4|pages=1188–1196|doi=10.1044/2024_PERSP-23-00251|issn=2381-4764}}</ref> Persons who are deaf or hard of hearing are particularly vulnerable during bombardments, as they are unable to hear warning sounds, explosions, or approaching danger, placing them at elevated risk of injury. The prolonged siege on Gaza and restrictions on medical supply entry have severely disrupted access to hearing aids, cochlear implant processors, batteries, spare parts, and routine maintenance services ([https://www.unicef.org/sop/stories/unicef-delivers-critical-hearing-aids-gaza-children UNICEF], 2025; [https://en.pngoportal.org/post/3906/In-Gaza-HearingLoss-a-Growing-Concern-Urgent-Audiology-Responses-in-Gaza-by-Atfaluna-Society PNGO portal], 2025). Many children and adults who previously relied on these devices have experienced extended periods without functioning equipment, resulting in auditory deprivation, regression in communication skills, and reduced participation in daily life. The impact extends beyond physiological hearing loss. Disrupted access to assistive devices and specialized care has significantly affected social interaction, emotional well-being, and access to education and psychosocial support, particularly for children with hearing disabilities. In both Gaza and the West Bank, intensified military incursions into refugee camps, villages, and urban neighborhoods—often involving sound grenades and live fire near civilian homes—have further increased the risk of noise-induced hearing damage. The ongoing prohibition on the entry of cochlear implant equipment and advanced audiological technologies, combined with a critical shortage of trained audiology professionals and medical infrastructure, continues to delay or prevent timely interventions, resulting in progressive and largely preventable deterioration of hearing health among affected populations ([https://www.atfaluna.org/en Atfaluna Society for Deaf Children]). {{HTitle|Audiology Charities}} Several NGOs and international partners support cochlear implant programs, hearing aid distribution, and community outreach initiatives in both regions. Despite these efforts, service availability remains insufficient, and many individuals wait years for hearing aids or cochlear implants due to financial hardship. * The [https://www.atfaluna.org/en Atfaluna Society for Deaf Children] (ASDC) in Gaza provides comprehensive audiology and speech-language services, including screening, diagnostic evaluation, hearing aid fitting, earmold fabrication, device repair, and speech therapy. * [https://en.wikipedia.org/wiki/UNICEF UNICEF] has supplied hundreds of hearing aids to Gaza, particularly during crises, with priority given to children with severe hearing loss. * The [https://en.wikipedia.org/wiki/Palestine_Red_Crescent_Society|Palestinian Red Crescent Society] provides hearing services in selected governorates, primarily through screening programs, hearing aid provision, and referrals for further audiological or ENT evaluation. {{HTitle|References}} {{reflist}} {{:Global Audiology/Authors-1|Layan Hamayel|https://www.researchgate.net/profile/Layan-Hamayel}} ''Edited by'' [https://orcid.org/0000-0002-4586-2398/ Nausheen Dawood] [[Category:Palestine]] [[Category:Audiology]] 3pmuiz7d6fz7h1m7wzlgziqmf5oib4p 2806761 2806752 2026-04-27T04:31:32Z RadiX 155307 fix 2806761 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Asia/Header}} {{CountryHeader|File:State of Palestine (orthographic projection).svg|https://en.wikipedia.org/wiki/Palestine}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Palestine Palestine], officially the State of [[Palestine]], is a partially recognized sovereign state in the Levant region of Western Asia. It encompasses the Israeli-occupied West Bank, including East Jerusalem, and the Gaza Strip, collectively known as the Palestinian territories. The territories share the vast majority of their borders with Israel, with the West Bank bordering Jordan to the east and the Gaza Strip bordering Egypt to the southwest. [[Arabic]] is the official language of the State of Palestine, specifically the Palestinian Arabic dialect. Hebrew and English are also widely spoken. {{HTitle|History of Audiology}} Audiology is a relatively new health profession in the West Bank and Gaza. The field has expanded gradually over the past three decades through university programs and the establishment of local clinics. Some hearing care services have also been supported through collaborations with international organizations, enabling the provision of hearing aids, cochlear implants, and otologic surgeries. Despite these developments, many towns and villages still lack dedicated audiology clinics, leaving large segments of the population underserved. {{HTitle|Incidence and Prevalence of Hearing Loss}} To date, no national epidemiological survey conducted by the [https://en.wikipedia.org/wiki/Ministry_of_Health_(Palestine)|Palestinian Ministry of Health] has comprehensively measured the incidence or prevalence of hearing loss across the West Bank and Gaza Strip ([https://moh.ps/mohsite/Content/Books/gaWxiN5DMkd4v4LhRW8YB5BqufSinMh9gJKgHIo6sS56PgPHOFb4ri%20mQCSoDDSxCysOckjMPMFbBJfuqS6dNU86jCVIpfehsjOVkQZrqDqZp.pdf Annual Health Report, 2024]). However, the only sustained newborn hearing screening initiative to date was established by [https://en.wikipedia.org/wiki/Caritas_Baby_Hospital Caritas Baby Hospital] in Bethlehem. Between September 25, 2006, and December 31, 2011, Caritas screened more than 15,000 newborns, reporting a higher-than-expected prevalence of infant hearing loss compared with global estimates, highlighting the urgent need for a national screening system.<ref>{{Cite journal|last=Corradin|first=Lucia|last2=Hindiyeh|first2=Musa|last3=Khaled|first3=Rasha|last4=Rishmawi|first4=Fadi|last5=Zidan|first5=Marwan|last6=Marzouqa|first6=Hiyam|date=2014-11-21|title=Survey on Infant Hearing Loss at Caritas Baby Hospital in Bethlehem-Palestine|url=https://www.mdpi.com/2039-4349/4/1/99|journal=Audiology Research|language=en|volume=4|issue=1|pages=99|doi=10.4081/audiores.2014.99|issn=2039-4349|pmc=4627132|pmid=26557353}}</ref> {{HTitle|Information About Audiology}} Hearing Care Services Professionals Audiology and hearing services are provided by audiologists, speech-language therapists, and ENT (otolaryngology) physicians across governmental, NGO, and private settings. Audiological Services * Diagnostic services: puretone audiometry, tympanometry, acoustic reflex testing, otoacoustic emissions (OAE), and auditory brainstem response (ABR). * Hearing aid services: fitting, programming, maintenance, repair, and earmold production. * Cochlear implant services: mapping, troubleshooting, and device maintenance. * Hearing rehabilitation: auditory training and speech-language therapy. * Vestibular assessment: limited availability of videonystagmography (VNG) in a few private clinics. ENT / Otologists / Otoneurologists ENT surgeons provide a range of surgical and medical services, including cochlear implantations and middle-ear surgeries. Some CI surgeries are conducted through collaborative or mission-based programs.<ref>{{Cite web|url=https://reliefweb.int/report/occupied-palestinian-territory/qrcs-provides-hearing-aids-gaza-newborn-children-enar|title=QRCS provides hearing aids for Gaza newborn children [EN/AR] - occupied Palestinian territory {{!}} ReliefWeb|date=2022-09-21|website=reliefweb.int|language=en|access-date=2025-12-13}}</ref> <ref>{{Cite web|url=https://alameen.ngo/announcing-the-opening-of-registration-for-the-auditory-rehabilitation-and-cochlear-implant-program-for-palestinian-children-residing-in-the-middle-east/|title=Announcing the opening of registration for the Auditory Rehabilitation and Cochlear Implant Program for Palestinian children residing in the Middle East|last=Nadeem|date=2024-06-26|website=IRVD|language=en-US|access-date=2025-12-13}}</ref> <ref>{{Cite web|url=https://spa.gov.sa/|title=KSrelief Volunteer Medical Team Carries out 40 Cochlear Implants for Palestinian Children|website=spa.gov.sa|language=en|access-date=2025-12-13}}</ref> <ref>{{Cite web|url=https://reliefweb.int/report/occupied-palestinian-territory/qrcs-provides-hearing-aids-gaza-newborn-children-enar|title=QRCS provides hearing aids for Gaza newborn children [EN/AR] - occupied Palestinian territory {{!}} ReliefWeb|date=2022-09-21|website=reliefweb.int|language=en|access-date=2025-12-13}}</ref> Services—such as balance and tinnitus evaluation—exist but remain limited and unevenly distributed. Primary Health Care Primary care centers refer individuals suspected to have hearing impairment; however, universal newborn hearing screening (UNHS) is still not implemented nationwide. There is no dedicated national law regulating audiology practice. Clinicians providing hearing care must hold a valid license, and all clinics require Ministry of Health approval. {{HTitle|Scope of Practice and Licensing}} Education and Professional Practice Most audiologists hold a Bachelor’s degree in Speech Therapy and Audiology, with some holding diplomas from institutions accredited by the Palestinian Ministry of Higher Education, including [https://en.wikipedia.org/wiki/An-Najah_National_University An-Najah National University], [https://en.wikipedia.org/wiki/Birzeit_University|Birzeit Birzeit University], [https://en.wikipedia.org/wiki/Bethlehem_University|Bethlehem Bethlehem University], [https://en.wikipedia.org/wiki/Arab%20American%20University%20(Palestine) Arab American University], [https://en.wikipedia.org/wiki/Al-Quds_University|Al-Quds Al-Quds University], and Al-Terra College. However, there are no fully specialized or advanced audiology degree programs (Master’s or PhD) currently offered. Professional and regulatory bodies Audiology and speech-language therapy are not yet overseen by a unified national regulatory body. Licensing is processed through the Palestinian Ministry of Health for graduates of approved programs. A new requirement introduced in October 2025 obligates newly graduated or previously unlicensed practitioners to pass a licensing examination. Audiologists’ scope of practice typically includes diagnostics, hearing aid services, and rehabilitation. Surgical interventions remain the domain of ENT specialists. {{HTitle|Professional and Regulatory Bodies}} There is no dedicated national research society for audiology. Research efforts are fragmented and largely driven by universities, hospitals, or individual graduate students. Only a small number of studies related to hearing and audiology have been conducted, and many have not been formally published. {{HTitle|Research in Audiology}} There is no dedicated national research society for audiology. Research efforts are fragmented and largely driven by universities, hospitals, or individual graduate students. Only a small number of studies related to hearing and audiology have been conducted, and many have not been formally published.<ref>{{Cite journal|last=El-Dabbakeh|first=Hatem S|date=2024-10-14|title=Risk Factors for Hearing Impairment Among Primary School Deaf Children in Gaza Strip, Palestine|url=https://journal.medtigo.com/risk-factors-for-hearing-impairment-among-primary-school-deaf-children-in-gaza-strip-palestine/|journal=medtigo Journal of Medicine|language=en|volume=1|issue=1|pages=1–5|doi=10.63096/medtigo30622423|issn=3066-3202}}</ref> <ref>{{Cite journal|last=Abu Rayyan|first=Amal|last2=Kamal|first2=Lara|last3=Casadei|first3=Silvia|last4=Brownstein|first4=Zippora|last5=Zahdeh|first5=Fouad|last6=Shahin|first6=Hashem|last7=Canavati|first7=Christina|last8=Dweik|first8=Dima|last9=Jaraysa|first9=Tamara|date=2020-08-18|title=Genomic analysis of inherited hearing loss in the Palestinian population|url=https://pnas.org/doi/full/10.1073/pnas.2009628117|journal=Proceedings of the National Academy of Sciences|language=en|volume=117|issue=33|pages=20070–20076|doi=10.1073/pnas.2009628117|issn=0027-8424|pmc=7443947|pmid=32747562}}</ref> {{HTitle|Challenges, Opportunities and Notes}} '''Challenges''' * Uneven geographic distribution of specialized services, with many remote areas underserved. * Heavy reliance on external funding for hearing aids and cochlear implants. * Absence of universal newborn hearing screening and standardized early detection protocols. * High genetic burden of hearing loss due to consanguinity, increasing the need for genetic counseling. * Growing number of audiology graduates versus a limited number of clinics. * Limited validated data and research on hearing health. * Lack of public awareness and early intervention advocacy campaigns. * Insufficient clinical training opportunities and hands-on practice for students and new graduates. * Weak regulatory oversight of clinics and professionals; some centers employ unqualified people. * Movement restrictions and import limitations affecting the access of hearing equipment and care. * Scarcity of specialists with advanced degrees in audiology. * Financial barriers for families seeking hearing aids or cochlear implants. * Social stigma surrounding hearing devices, contributing to delayed intervention and detection. '''Opportunities''' * New academic programs in audiology at Palestinian universities open the way for more local professionals * Collaboration with NGOs and international partners to implement newborn hearing screening and protocols * Development of a national regulatory body to standardize practice and licensing. * Strengthening Palestine Ministry of Health oversight and accountability. * Establishing additional hearing care clinics through local or international partnerships. {{HTitle|Impact of the war on hearing loss}} Prolonged armed conflict, repeated military incursions in the West Bank, and the ongoing war in the Gaza Strip have had profound and cumulative effects on hearing health, particularly among persons with hearing disabilities. Continuous exposure to high intensity noise sources—such as airstrikes, artillery fire, sound bombs, live ammunition, and explosions—has been linked to acoustic trauma, tinnitus, and temporary or permanent hearing loss among both children and adults [UN News, 2024]. In Gaza, the low flying military aircraft, drones, and surveillance planes that operate continuously, sometimes for 24 hours a day, have been argued to contribute to persistent tinnitus, auditory fatigue, sleep disturbances, and stress related auditory symptoms. Recent local reports estimated that as many as 35,000 individuals in Gaza are experience hearing problems attributable to the extreme exposure from bombings, underscoring the severity of war-related auditory impacts on civilians. Reports from international and local organizations indicate that the documented prevalence of hearing loss in Gaza and the West Bank significantly underestimates the true burden, as access to audiological assessment, diagnosis, and rehabilitation services is extremely limited or entirely unavailable during periods of active conflict.<ref>{{Cite journal|last=Pakulski|first=Lori A.|date=2024-08|title=Addressing Hearing Loss of Palestinians Living in Refugee Camps|url=https://pubs.asha.org/doi/10.1044/2024_PERSP-23-00251|journal=Perspectives of the ASHA Special Interest Groups|language=en|volume=9|issue=4|pages=1188–1196|doi=10.1044/2024_PERSP-23-00251|issn=2381-4764}}</ref> Persons who are deaf or hard of hearing are particularly vulnerable during bombardments, as they are unable to hear warning sounds, explosions, or approaching danger, placing them at elevated risk of injury. The prolonged siege on Gaza and restrictions on medical supply entry have severely disrupted access to hearing aids, cochlear implant processors, batteries, spare parts, and routine maintenance services ([https://www.unicef.org/sop/stories/unicef-delivers-critical-hearing-aids-gaza-children UNICEF], 2025; [https://en.pngoportal.org/post/3906/In-Gaza-HearingLoss-a-Growing-Concern-Urgent-Audiology-Responses-in-Gaza-by-Atfaluna-Society PNGO portal], 2025). Many children and adults who previously relied on these devices have experienced extended periods without functioning equipment, resulting in auditory deprivation, regression in communication skills, and reduced participation in daily life. The impact extends beyond physiological hearing loss. Disrupted access to assistive devices and specialized care has significantly affected social interaction, emotional well-being, and access to education and psychosocial support, particularly for children with hearing disabilities. In both Gaza and the West Bank, intensified military incursions into refugee camps, villages, and urban neighborhoods—often involving sound grenades and live fire near civilian homes—have further increased the risk of noise-induced hearing damage. The ongoing prohibition on the entry of cochlear implant equipment and advanced audiological technologies, combined with a critical shortage of trained audiology professionals and medical infrastructure, continues to delay or prevent timely interventions, resulting in progressive and largely preventable deterioration of hearing health among affected populations ([https://www.atfaluna.org/en Atfaluna Society for Deaf Children]). {{HTitle|Audiology Charities}} Several NGOs and international partners support cochlear implant programs, hearing aid distribution, and community outreach initiatives in both regions. Despite these efforts, service availability remains insufficient, and many individuals wait years for hearing aids or cochlear implants due to financial hardship. * The [https://www.atfaluna.org/en Atfaluna Society for Deaf Children] (ASDC) in Gaza provides comprehensive audiology and speech-language services, including screening, diagnostic evaluation, hearing aid fitting, earmold fabrication, device repair, and speech therapy. * [https://en.wikipedia.org/wiki/UNICEF UNICEF] has supplied hundreds of hearing aids to Gaza, particularly during crises, with priority given to children with severe hearing loss. * The [https://en.wikipedia.org/wiki/Palestine_Red_Crescent_Society|Palestinian Red Crescent Society] provides hearing services in selected governorates, primarily through screening programs, hearing aid provision, and referrals for further audiological or ENT evaluation. {{HTitle|References}} {{reflist}} {{Global Audiology Authors |name1=Layan Hamayel |role1=Author |researchgate1=https://www.researchgate.net/profile/Layan-Hamayel |name2=Nausheen Dawood |role2=Contributor |orcid2=https://orcid.org/0000-0002-4586-2398/ }} [[Category:Palestine]] [[Category:Audiology]] b5jlohyrjqlj3p4fbkcoagsmk8kn1bv 2 326765 2806685 2806510 2026-04-26T18:18:31Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2806685 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct chord lengths of the unit-radius 120-cell, the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius coordinate system, the original planar octagon we started with has chords of length: :<math>r_1=\sqrt{2},r_2==\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} fl72f97gl54ksz5f6znx0apktwn08vi 2806686 2806685 2026-04-26T18:20:47Z Dc.samizdat 2856930 /* Thirty distinguished distances */ 2806686 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example, there is a section of the 120-cell which is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius coordinate system, the original planar octagon we started with has chords of length: :<math>r_1=\sqrt{2},r_2==\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} mgt9tfr4rsc9v7c9oxpxt81f7bjxz97 2806687 2806686 2026-04-26T18:32:40Z Dc.samizdat 2856930 /* Visualizing the 120-cell */ 2806687 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius coordinate system, the original planar octagon we started with has chords of length: :<math>r_1=\sqrt{2},r_2==\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} qivlgkmddzwl0a5rz47oqzf2q9vs2hr 2806692 2806687 2026-04-26T18:46:18Z Dc.samizdat 2856930 /* Visualizing the 120-cell */ 2806692 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:Stellation core|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius coordinate system, the original planar octagon we started with has chords of length: :<math>r_1=\sqrt{2},r_2==\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} rsmh28yjykd6egzbrr5vj1m5nfute1u 2806694 2806692 2026-04-26T19:14:11Z Dc.samizdat 2856930 /* Compounds in the 120-cell */ 2806694 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius coordinate system, the original planar octagon we started with has chords of length: :<math>r_1=\sqrt{2},r_2==\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} mximvrfcf4exhbfyb7b8q3oxsbz2zqw 2806712 2806694 2026-04-26T21:53:55Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2806712 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius coordinate system, the original planar octagon we started with has chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> The vertex coordinates form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} se1fn03w48ev9umlqo7c3jwwtc45ies 2806715 2806712 2026-04-26T22:06:13Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2806715 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords, except <math>r_1=\sqrt{2}</math>, occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 3h869ah09brp8c623kyic74qknqhe2y 2806716 2806715 2026-04-26T22:07:57Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2806716 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} t91stvl3r4647p0tiyi1jqvdegrtabm 2806721 2806716 2026-04-26T22:34:40Z Dc.samizdat 2856930 /* Hypercubes */ 2806721 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The 4-hypercube [[tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells. It is the four-dimensional analogue of the cube. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} pyittscjdb74jsfyosgz5dqk7gu85hu 2806722 2806721 2026-04-26T22:35:03Z Dc.samizdat 2856930 /* Hypercubes */ 2806722 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The 4-hypercube [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells. It is the four-dimensional analogue of the cube. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} hkzdr5rr6tac5d2qb2z4dox0ndl7xvj 2806724 2806722 2026-04-26T22:37:31Z Dc.samizdat 2856930 /* Hypercubes */ 2806724 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The 4-hypercube [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} eq3f5p2ziiu01ehowvza4sxcl7d21gl 2806725 2806724 2026-04-26T22:39:32Z Dc.samizdat 2856930 /* Hypercubes */ 2806725 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. ... == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} nx92w0oqutny6xzwlw844eo08s94ead 2806727 2806725 2026-04-26T23:12:40Z Dc.samizdat 2856930 /* Hypercubes */ 2806727 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the dual polytope of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, has 2 disjoint instances and 6 distinct instances of its Petrie polygon. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} k57oz5y6tzyudn7c19qzm4xxt6gp6ig 2806728 2806727 2026-04-27T00:57:36Z Dc.samizdat 2856930 /* Hypercubes */ 2806728 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. ... == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the dual polytope of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the unit-radius tesseract is radially equilateral (unlike the 16-cell), to build it we can start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 1ydk58ilq7jumxfoagepl9oxn17tvt0 2806729 2806728 2026-04-27T00:58:05Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2806729 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the dual polytope of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the unit-radius tesseract is radially equilateral (unlike the 16-cell), to build it we can start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 0wt8egxiqncly7gno3hnr3kk3mzq1pg 2806730 2806729 2026-04-27T01:11:39Z Dc.samizdat 2856930 /* Hypercubes */ 2806730 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the 8-point cube, which is the convex hull of a [[Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the unit-radius tesseract is radially equilateral (unlike the 16-cell), to build it we can start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 6pgld479x1c4k4cp8izj9bmziyvy3er 2806731 2806730 2026-04-27T01:12:04Z Dc.samizdat 2856930 /* Hypercubes */ 2806731 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the 8-point cube, which is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the unit-radius tesseract is radially equilateral (unlike the 16-cell), to build it we can start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} t93gc54q5d89l4dr1pvrbqfykaxewa7 2806732 2806731 2026-04-27T01:12:52Z Dc.samizdat 2856930 /* Hypercubes */ 2806732 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the unit-radius tesseract is radially equilateral (unlike the 16-cell), to build it we can start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} myaogkxnq8j71qigt6urthaw42w8sh2 2806733 2806732 2026-04-27T01:14:39Z Dc.samizdat 2856930 /* Hypercubes */ 2806733 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the tesseract is radially equilateral (unlike the 16-cell), to build it we can start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} kx4hjs5zhjonsk9uqh9bwgm98ydebda 2806734 2806733 2026-04-27T01:20:15Z Dc.samizdat 2856930 /* The 24-cell */ 2806734 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract, with twice as many vertices, contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 6h0y4lvc04pviu9vmpcm3t645sx6hbf 2806735 2806734 2026-04-27T01:46:40Z Dc.samizdat 2856930 /* Hypercubes */ 2806735 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge-length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge-length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} bnegbacw1ck7xyrkde7saei5sq8a8so 2806736 2806735 2026-04-27T01:52:01Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2806736 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} gj9hnp845bro5i1yukssoi95qokx5w8 2806769 2806736 2026-04-27T06:11:51Z Dc.samizdat 2856930 /* Hypercubes */ 2806769 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and every vertex to its antipodal vertex 180° degrees away. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} kn7eolnvxijxch54s43dtvnaa6hbqf1 0 327333 2806746 2804987 2026-04-27T04:11:04Z RadiX 155307 2806746 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Cameroon (orthographic projection).svg|https://en.wikipedia.org/wiki/Cameroon}} {{HTitle|General Information}} '''[https://en.wikipedia.org/wiki/Cameroon Cameroon]''', officially the Republic of Cameroon, is a country in Central Africa. It shares boundaries with Nigeria to the west and north, Chad to the northeast, the Central African Republic to the east, and Equatorial Guinea, Gabon, and the Republic of the Congo to the south. In addition to numerous traditional languages, the two official national languages, French (80%) and English (20%), reflect the now sovereign republic’s history of governance by France and England after WWI. {{HTitle|History of Audiology}} While there is a federally run public healthcare system, access to services and quality of care vary greatly by region. The general health situation in the insecure areas of Cameroon is characterized by a significant decline in the services provided by qualified health personnels. This is exasperated by a general lack of preventive medical and rehabilitative measures specifically with specialty services, like ear and hearing. People with hearing impairment are therefore particularly vulnerable and face social and economic barriers to care. ==== Hearing Care Services: ==== Ear and hearing care in Cameroon developed slowly and for many decades existed only as part of general ENT (Ear, Nose, and Throat) services in major hospitals. During the colonial and early post-independence period (1960s–1990s), care focused mainly on treating ear infections and ENT conditions, with no structured audiology services, hearing screening, or national policies. From the early 2000s, limited specialist ENT services became available in urban referral hospitals, but access to hearing assessment, hearing aids, and rehabilitation remained extremely limited. Not only were there very few trained ENT specialists but also most ENTs in Cameroon focus primarily on nose and throat care and almost no clinicians providing audiology services, especially outside major cities. A major shift occurred in the 2010s, driven largely by NGO and faith-based organizations, particularly the Cameroon Baptist Convention Health Services (CBCHS)[https://en.wikipedia.org/wiki/CBM_(charity)] with support from Christian Blind Mission (CBM)[https://www.cbm.org/news/news-articles/2025/BMZ-Funded-Project-on-Ear-Diseases-and-Hearing-Loss-in-Cameroon.html] of Germany. These organizations conducted situational analyses that highlighted large gaps in workforce, equipment, early detection, and policy. ENT providers most provide nose and throat focused service and there is no licensed profession of Audiology. Around 2020, Cameroon saw its first organized ear and hearing care initiatives, including pilot neonatal hearing screening, basic audiology services, community awareness, and training of primary healthcare workers.[https://cbchealthservices.org/cbchs-cbm-pilot-audiology-training-in-cameron/] Advanced services such as cochlear implantation were introduced on a very limited scale through international collaboration. This initiative was carried out by the Cameroon Baptist Convention Health.[https://cbchealthservices.org/] From 2021 to the present, ear and hearing care has expanded through donor-funded NGO advocacy and national capacity building initiatives. This current period marks the beginning of efforts to develop a national ear and hearing care strategy, integrate services into primary healthcare, and improve early detection and rehabilitation. Overall, ear and hearing care in Cameroon has evolved from ENT-based treatment to early-stage system development, with growing recognition at national level but ongoing challenges in workforce, access, funding, and nationwide coverage. {{HTitle|Incidence and Prevalence of Hearing Loss}} Population based studies report the incidence of hearing impairment in Cameroon is between 0.9% - 3.6% of the general population depending on study quality, clinical tests, and diagnostic criteria utilized.<ref>{{Cite journal|title=Hearing Impairment Overview in Africa: the Case of Cameroon|url=https://pubmed.ncbi.nlm.nih.gov/32098311|journal=Genes|date=2020-02-22|issn=2073-4425|pmc=7073999|pmid=32098311|pages=233|volume=11|issue=2|doi=10.3390/genes11020233|first=Edmond|last=Wonkam Tingang|first2=Jean Jacques|last2=Noubiap|first3=Jean Valentin|last3=F Fokouo|first4=Oluwafemi Gabriel|last4=Oluwole|first5=Séraphin|last5=Nguefack|first6=Emile R.|last6=Chimusa|first7=Ambroise|last7=Wonkam}}</ref> Evaluations of childhood hearing loss in the country found that the most common type of loss identified is sensorineural, the most common degree/severity of loss is moderate, and reports of bilateral versus unilateral losses vary drastically between reports. The greatest cause of hearing loss in Cameroon is environment factors (52.6-62.2%) lead by vaccine preventable disease such as meningitis and chronic otitis media and genetic factors account for .8-14.8% though studies report 32.6-37% of cases have unknown origin. Overall, results vary drastically lessening the reliability of data and indicating a need for more standardized evaluation protocols across ear and hearing studies. {{HTitle|Information About Audiology}} ==== Audiological services ==== Overall, services vary by region. Urban areas (especially Yaoundé and Douala) have better access to diagnostics and hearing aids because they hav the most ENT specialists and private hearing centers. Whereas rural and remote areas rely mainly on primary health centers, district hospitals with very limited ear care, and NGO outreach services when available. The availability of treatment and technology is a major challenge in Cameroon. Available services can be classified as screening, diagnosis, or treatment/rehabilitation. * Screening. Community and clinic-based hearing screening for adults and children is being offered through campaigns and health events, often linked to World Hearing Day and local health outreach activities. Neonatal (newborn) hearing screening has been introduced in some hospitals, with thousands of babies screened using Otoacoustic Emissions (OAE) machines as part of pilot programmes. There are only two facilities that do ABR evaluations if a baby fails two OAE screenings. * Diagnosis and Clinical Services. ENT services at regional and district hospitals provide basic diagnosis and treatment of ear diseases and hearing problems. Audiology assessment and training are currently being developed through new training programmes to build local capacity for proper hearing diagnostics and referrals. * Treatment and Rehabilitation. Private centers and service providers offer hearing aid fitting, sales, and basic services in cities like Yaoundé. National hearing aid dispensing programs are emerging, enabling procurement and fitting of hearing aids for people with hearing loss. {{HTitle|Scope of Practice and Licensing}} ==== Professionals providing hearing care services ==== The current ear and hearing care professionals include; ENT physicians who conduct surgeries, ENT clinicians (primary ear and hearing care nurses) that assess, refer, and conduct minor interventions such as antibiotic prescriptions, ear and hearing care clinical officers who carry out early interventions and management, and audiology technicians (trained via NGO programs) who are able to carry out more advanced tests such as pure tone audiometry and hearing aid fittings. Primary health care providers play a role in early identification and referral to the hospitals. They may also carry out basic procedures such as removal of foreign bodies and ear washing. Ear and hearing care services in Cameroon vary in level of care, availability, and the type of facilities offering care: * Tertiary / Referral Hospitals, General Hospitals (e.g. Yaoundé, Douala), Central and Teaching Hospitals. Services: ENT consultations, diagnosis, ear surgery, limited audiology tests (ex. ABR), rare cochlear implant procedures as facilitated by NGOs (availability is mainly urban). * Private Regional Hospitals, Public regional hospitals. Services: ENT clinics, ear disease treatment, basic hearing assessment (availability: selected regions). * District Hospitals, Government district hospitals Services: Basic ear care, referrals, occasional hearing screening (availability: widespread but of limited scope). * Mission and Faith-Based Hospitals, Cameroon Baptist Convention Health Services (CBCHS) facilities, other church-run hospitals. Services: ENT care, hearing screening (including newborn screening in some sites), surgeries, hearing aids, community outreach programs (availability: multiple regions; key providers of hearing care). * Primary Health Care Facilities, Integrated Health Centres (IHCs), Health posts. Services: Basic ear care, health education, early identification, ENT referrals (availability: nationwide but very basic services). * Private Clinics and Hearing Centres, Private ENT clinics, Audiology/hearing aid centres. Services: Hearing tests, hearing aid fitting and sales (availability: only major cities). * NGOs and Community-Based Organisations, Disability and hearing-focused NGOs. Services: Community screening, rehabilitation support, awareness, referrals, sign language support (availability: Project-based, selected regions). * Schools for the Deaf / Special Education Centres, Special schools and inclusive education settings. Services: Communication support, identification of hearing loss, referrals (availability: Limited number nationwide, vary based on language). {{HTitle|Professional and Regulatory Bodies}} ==== Laws related to hearing care services ==== There is no specific law in Cameroon that directly governs ear and hearing care services. However, existing disability and education laws provide a legal basis for the rights of people with hearing loss to access services, education and protection. With the advent and implementation of the "Strengthening ear and hearing care capacities in Cameroon" project, implemented by the Cameroon Baptist Convention Health Services in partnership with the Christian Blind Mission and funding from the German Ministry for Economic Development and Cooperation, a national plan for ear and hearing health is set to be developed by 2028. ==== Education and Professional Practice ==== Cameroon currently has no formal university degree program dedicated specifically to audiology. Training for ear and hearing health professionals is mainly short-term, project-based, or embedded within other health disciplines. Available training includes pilot audiology technician programs and short courses organized by NGOs and faith-based institutions (notably CBCHS with CBM support). These focus on basic hearing screening, ear examination, referral, and limited hearing-aid fitting. Workshops and in-service training are also provided for nurses, primary health workers, and clinicians to strengthen basic ear care and early identification of hearing problems. Audiology is not yet formally regulated in Cameroon, and therefore there is no audiology defined education criteria, regulatory body, of defined scope of practice. Research in the ear and hearing field has been conducted primarily by ENT physicians and covers the incidence of disease and prevalence of hearing loss in country. {{HTitle|Ongoing audiology research}} Recently, research on hearing care in Cameroon has focused on addressing diagnostic practices and the challenges faced by individuals with hearing impairments. Key areas of focus include: * Diagnostic practices, attitudes and equipment availability. <ref>{{Cite journal|last=Choffor-Nchinda|first=Emmanuel|last2=Fokouo Fogha|first2=Jean Valentin|last3=Ngo Nyeki|first3=Adèle-Rose|last4=Dalil|first4=Asmaou Bouba|last5=Meva’a Biouélé|first5=Roger Christian|last6=Me-Meke|first6=Geschiere Peter|date=2022-12|title=Approach and solutions to congenital hearing impairment in Cameroon: perspective of hearing professionals|url=https://tropmedhealth.biomedcentral.com/articles/10.1186/s41182-022-00430-7|journal=Tropical Medicine and Health|language=en|volume=50|issue=1|doi=10.1186/s41182-022-00430-7|issn=1349-4147|pmc=9150302|pmid=35637511}}</ref> * Etiological profile of childhood deafness.<ref>{{Cite journal|last=Wonkam|first=Ambroise|last2=Noubiap|first2=Jean Jacques N.|last3=Djomou|first3=François|last4=Fieggen|first4=Karen|last5=Njock|first5=Richard|last6=Toure|first6=Geneviève Bengono|date=2013-01|title=Aetiology of childhood hearing loss in Cameroon (sub-Saharan Africa)|url=https://linkinghub.elsevier.com/retrieve/pii/S1769721212002777|journal=European Journal of Medical Genetics|language=en|volume=56|issue=1|pages=20–25|doi=10.1016/j.ejmg.2012.09.010}}</ref> * Prevalence and causes of hearing impairment. <ref>{{Cite journal|last=Ferrite|first=Silvia|last2=Mactaggart|first2=Islay|last3=Kuper|first3=Hannah|last4=Oye|first4=Joseph|last5=Polack|first5=Sarah|date=2017-04|title=Prevalence and causes of hearing impairment in Fundong Health District, North‐West Cameroon|url=https://onlinelibrary.wiley.com/doi/10.1111/tmi.12840|journal=Tropical Medicine & International Health|language=en|volume=22|issue=4|pages=485–492|doi=10.1111/tmi.12840|issn=1360-2276}}</ref> * Social and healthcare challenges and stigma.<ref>{{Cite journal|last=Wonkam-Tingang|first=Edmond|last2=Kengne Kamga|first2=Karen|last3=Adadey|first3=Samuel Mawuli|last4=Nguefack|first4=Seraphin|last5=De Kock|first5=Carmen|last6=Munung|first6=Nchangwi Syntia|last7=Wonkam|first7=Ambroise|date=2021-11-18|title=Knowledge and Challenges Associated With Hearing Impairment in Affected Individuals From Cameroon (Sub-Saharan Africa)|url=https://www.frontiersin.org/articles/10.3389/fresc.2021.726761/full|journal=Frontiers in Rehabilitation Sciences|volume=2|doi=10.3389/fresc.2021.726761|issn=2673-6861|pmc=9397862|pmid=36188771}}</ref> {{HTitle|Challenges, Opportunities and Notes}} "The Strengthening ear and hearing care capacities in Cameroon" project, implemented by the [https://en.wikipedia.org/wiki/Cameroon%20Baptist%20Convention Cameroon Baptist Convention Health Services] in partnership with the [https://en.wikipedia.org/wiki/CBM_(charity) Christian Blind Mission] and funding from the German Ministry for Economic Development and Cooperation. The aim of the planned project is to improve the quality of life of people with hearing loss or at risk of hearing loss in the following target regions: Northwest, Centre, Littoral. The project will focus on three areas of intervention: Capacity building of existing health professionals through continuing education, access to consulting and treatment services and, development of a national plan to strengthen ear and hearing health care. The aim of the planned project is to improve the quality of life of people with hearing loss or at risk of hearing loss in the following target regions: Northwest, Centre, Littoral. {{HTitle|Audiology Charities}} * The [https://www.bmz.de/en/ German Federal Ministry for Economic Cooperation and Development] (BMZ) * [https://www.cbm.org/ Christian Blind Mission] (CBM) * [https://cbchealthservices.org/ Cameroon Baptist Convention (CBC) Health Services] * [https://terptree.co.uk/communication-professionals/who-are-sound-seekers/ Sound Seekers] (UK) {{HTitle|References}} {{reflist}} {{:Global Audiology/Authors-1|Aisha Kinyuy|https://www.linkedin.com/in/aisha-kinyuy-7278471b5/}} ''Edited by'' [https://www.linkedin.com/in/mcmsaunders/ Madison Saunders] [[Category:Audiology]] [[Category:Cameroon]] 9rm17zbo3vyc3nu5frvyjipsgetcgzo 2806766 2806746 2026-04-27T04:41:16Z RadiX 155307 upd 2806766 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Africa/Header}} {{CountryHeader|File:Cameroon (orthographic projection).svg|https://en.wikipedia.org/wiki/Cameroon}} {{HTitle|General Information}} '''[https://en.wikipedia.org/wiki/Cameroon Cameroon]''', officially the Republic of Cameroon, is a country in Central Africa. It shares boundaries with Nigeria to the west and north, Chad to the northeast, the Central African Republic to the east, and Equatorial Guinea, Gabon, and the Republic of the Congo to the south. In addition to numerous traditional languages, the two official national languages, French (80%) and English (20%), reflect the now sovereign republic’s history of governance by France and England after WWI. {{HTitle|History of Audiology}} While there is a federally run public healthcare system, access to services and quality of care vary greatly by region. The general health situation in the insecure areas of Cameroon is characterized by a significant decline in the services provided by qualified health personnels. This is exasperated by a general lack of preventive medical and rehabilitative measures specifically with specialty services, like ear and hearing. People with hearing impairment are therefore particularly vulnerable and face social and economic barriers to care. ==== Hearing Care Services: ==== Ear and hearing care in Cameroon developed slowly and for many decades existed only as part of general ENT (Ear, Nose, and Throat) services in major hospitals. During the colonial and early post-independence period (1960s–1990s), care focused mainly on treating ear infections and ENT conditions, with no structured audiology services, hearing screening, or national policies. From the early 2000s, limited specialist ENT services became available in urban referral hospitals, but access to hearing assessment, hearing aids, and rehabilitation remained extremely limited. Not only were there very few trained ENT specialists but also most ENTs in Cameroon focus primarily on nose and throat care and almost no clinicians providing audiology services, especially outside major cities. A major shift occurred in the 2010s, driven largely by NGO and faith-based organizations, particularly the Cameroon Baptist Convention Health Services (CBCHS)[https://en.wikipedia.org/wiki/CBM_(charity)] with support from Christian Blind Mission (CBM)[https://www.cbm.org/news/news-articles/2025/BMZ-Funded-Project-on-Ear-Diseases-and-Hearing-Loss-in-Cameroon.html] of Germany. These organizations conducted situational analyses that highlighted large gaps in workforce, equipment, early detection, and policy. ENT providers most provide nose and throat focused service and there is no licensed profession of Audiology. Around 2020, Cameroon saw its first organized ear and hearing care initiatives, including pilot neonatal hearing screening, basic audiology services, community awareness, and training of primary healthcare workers.[https://cbchealthservices.org/cbchs-cbm-pilot-audiology-training-in-cameron/] Advanced services such as cochlear implantation were introduced on a very limited scale through international collaboration. This initiative was carried out by the Cameroon Baptist Convention Health.[https://cbchealthservices.org/] From 2021 to the present, ear and hearing care has expanded through donor-funded NGO advocacy and national capacity building initiatives. This current period marks the beginning of efforts to develop a national ear and hearing care strategy, integrate services into primary healthcare, and improve early detection and rehabilitation. Overall, ear and hearing care in Cameroon has evolved from ENT-based treatment to early-stage system development, with growing recognition at national level but ongoing challenges in workforce, access, funding, and nationwide coverage. {{HTitle|Incidence and Prevalence of Hearing Loss}} Population based studies report the incidence of hearing impairment in Cameroon is between 0.9% - 3.6% of the general population depending on study quality, clinical tests, and diagnostic criteria utilized.<ref>{{Cite journal|title=Hearing Impairment Overview in Africa: the Case of Cameroon|url=https://pubmed.ncbi.nlm.nih.gov/32098311|journal=Genes|date=2020-02-22|issn=2073-4425|pmc=7073999|pmid=32098311|pages=233|volume=11|issue=2|doi=10.3390/genes11020233|first=Edmond|last=Wonkam Tingang|first2=Jean Jacques|last2=Noubiap|first3=Jean Valentin|last3=F Fokouo|first4=Oluwafemi Gabriel|last4=Oluwole|first5=Séraphin|last5=Nguefack|first6=Emile R.|last6=Chimusa|first7=Ambroise|last7=Wonkam}}</ref> Evaluations of childhood hearing loss in the country found that the most common type of loss identified is sensorineural, the most common degree/severity of loss is moderate, and reports of bilateral versus unilateral losses vary drastically between reports. The greatest cause of hearing loss in Cameroon is environment factors (52.6-62.2%) lead by vaccine preventable disease such as meningitis and chronic otitis media and genetic factors account for .8-14.8% though studies report 32.6-37% of cases have unknown origin. Overall, results vary drastically lessening the reliability of data and indicating a need for more standardized evaluation protocols across ear and hearing studies. {{HTitle|Information About Audiology}} ==== Audiological services ==== Overall, services vary by region. Urban areas (especially Yaoundé and Douala) have better access to diagnostics and hearing aids because they hav the most ENT specialists and private hearing centers. Whereas rural and remote areas rely mainly on primary health centers, district hospitals with very limited ear care, and NGO outreach services when available. The availability of treatment and technology is a major challenge in Cameroon. Available services can be classified as screening, diagnosis, or treatment/rehabilitation. * Screening. Community and clinic-based hearing screening for adults and children is being offered through campaigns and health events, often linked to World Hearing Day and local health outreach activities. Neonatal (newborn) hearing screening has been introduced in some hospitals, with thousands of babies screened using Otoacoustic Emissions (OAE) machines as part of pilot programmes. There are only two facilities that do ABR evaluations if a baby fails two OAE screenings. * Diagnosis and Clinical Services. ENT services at regional and district hospitals provide basic diagnosis and treatment of ear diseases and hearing problems. Audiology assessment and training are currently being developed through new training programmes to build local capacity for proper hearing diagnostics and referrals. * Treatment and Rehabilitation. Private centers and service providers offer hearing aid fitting, sales, and basic services in cities like Yaoundé. National hearing aid dispensing programs are emerging, enabling procurement and fitting of hearing aids for people with hearing loss. {{HTitle|Scope of Practice and Licensing}} ==== Professionals providing hearing care services ==== The current ear and hearing care professionals include; ENT physicians who conduct surgeries, ENT clinicians (primary ear and hearing care nurses) that assess, refer, and conduct minor interventions such as antibiotic prescriptions, ear and hearing care clinical officers who carry out early interventions and management, and audiology technicians (trained via NGO programs) who are able to carry out more advanced tests such as pure tone audiometry and hearing aid fittings. Primary health care providers play a role in early identification and referral to the hospitals. They may also carry out basic procedures such as removal of foreign bodies and ear washing. Ear and hearing care services in Cameroon vary in level of care, availability, and the type of facilities offering care: * Tertiary / Referral Hospitals, General Hospitals (e.g. Yaoundé, Douala), Central and Teaching Hospitals. Services: ENT consultations, diagnosis, ear surgery, limited audiology tests (ex. ABR), rare cochlear implant procedures as facilitated by NGOs (availability is mainly urban). * Private Regional Hospitals, Public regional hospitals. Services: ENT clinics, ear disease treatment, basic hearing assessment (availability: selected regions). * District Hospitals, Government district hospitals Services: Basic ear care, referrals, occasional hearing screening (availability: widespread but of limited scope). * Mission and Faith-Based Hospitals, Cameroon Baptist Convention Health Services (CBCHS) facilities, other church-run hospitals. Services: ENT care, hearing screening (including newborn screening in some sites), surgeries, hearing aids, community outreach programs (availability: multiple regions; key providers of hearing care). * Primary Health Care Facilities, Integrated Health Centres (IHCs), Health posts. Services: Basic ear care, health education, early identification, ENT referrals (availability: nationwide but very basic services). * Private Clinics and Hearing Centres, Private ENT clinics, Audiology/hearing aid centres. Services: Hearing tests, hearing aid fitting and sales (availability: only major cities). * NGOs and Community-Based Organisations, Disability and hearing-focused NGOs. Services: Community screening, rehabilitation support, awareness, referrals, sign language support (availability: Project-based, selected regions). * Schools for the Deaf / Special Education Centres, Special schools and inclusive education settings. Services: Communication support, identification of hearing loss, referrals (availability: Limited number nationwide, vary based on language). {{HTitle|Professional and Regulatory Bodies}} ==== Laws related to hearing care services ==== There is no specific law in Cameroon that directly governs ear and hearing care services. However, existing disability and education laws provide a legal basis for the rights of people with hearing loss to access services, education and protection. With the advent and implementation of the "Strengthening ear and hearing care capacities in Cameroon" project, implemented by the Cameroon Baptist Convention Health Services in partnership with the Christian Blind Mission and funding from the German Ministry for Economic Development and Cooperation, a national plan for ear and hearing health is set to be developed by 2028. ==== Education and Professional Practice ==== Cameroon currently has no formal university degree program dedicated specifically to audiology. Training for ear and hearing health professionals is mainly short-term, project-based, or embedded within other health disciplines. Available training includes pilot audiology technician programs and short courses organized by NGOs and faith-based institutions (notably CBCHS with CBM support). These focus on basic hearing screening, ear examination, referral, and limited hearing-aid fitting. Workshops and in-service training are also provided for nurses, primary health workers, and clinicians to strengthen basic ear care and early identification of hearing problems. Audiology is not yet formally regulated in Cameroon, and therefore there is no audiology defined education criteria, regulatory body, of defined scope of practice. Research in the ear and hearing field has been conducted primarily by ENT physicians and covers the incidence of disease and prevalence of hearing loss in country. {{HTitle|Ongoing audiology research}} Recently, research on hearing care in Cameroon has focused on addressing diagnostic practices and the challenges faced by individuals with hearing impairments. Key areas of focus include: * Diagnostic practices, attitudes and equipment availability. <ref>{{Cite journal|last=Choffor-Nchinda|first=Emmanuel|last2=Fokouo Fogha|first2=Jean Valentin|last3=Ngo Nyeki|first3=Adèle-Rose|last4=Dalil|first4=Asmaou Bouba|last5=Meva’a Biouélé|first5=Roger Christian|last6=Me-Meke|first6=Geschiere Peter|date=2022-12|title=Approach and solutions to congenital hearing impairment in Cameroon: perspective of hearing professionals|url=https://tropmedhealth.biomedcentral.com/articles/10.1186/s41182-022-00430-7|journal=Tropical Medicine and Health|language=en|volume=50|issue=1|doi=10.1186/s41182-022-00430-7|issn=1349-4147|pmc=9150302|pmid=35637511}}</ref> * Etiological profile of childhood deafness.<ref>{{Cite journal|last=Wonkam|first=Ambroise|last2=Noubiap|first2=Jean Jacques N.|last3=Djomou|first3=François|last4=Fieggen|first4=Karen|last5=Njock|first5=Richard|last6=Toure|first6=Geneviève Bengono|date=2013-01|title=Aetiology of childhood hearing loss in Cameroon (sub-Saharan Africa)|url=https://linkinghub.elsevier.com/retrieve/pii/S1769721212002777|journal=European Journal of Medical Genetics|language=en|volume=56|issue=1|pages=20–25|doi=10.1016/j.ejmg.2012.09.010}}</ref> * Prevalence and causes of hearing impairment. <ref>{{Cite journal|last=Ferrite|first=Silvia|last2=Mactaggart|first2=Islay|last3=Kuper|first3=Hannah|last4=Oye|first4=Joseph|last5=Polack|first5=Sarah|date=2017-04|title=Prevalence and causes of hearing impairment in Fundong Health District, North‐West Cameroon|url=https://onlinelibrary.wiley.com/doi/10.1111/tmi.12840|journal=Tropical Medicine & International Health|language=en|volume=22|issue=4|pages=485–492|doi=10.1111/tmi.12840|issn=1360-2276}}</ref> * Social and healthcare challenges and stigma.<ref>{{Cite journal|last=Wonkam-Tingang|first=Edmond|last2=Kengne Kamga|first2=Karen|last3=Adadey|first3=Samuel Mawuli|last4=Nguefack|first4=Seraphin|last5=De Kock|first5=Carmen|last6=Munung|first6=Nchangwi Syntia|last7=Wonkam|first7=Ambroise|date=2021-11-18|title=Knowledge and Challenges Associated With Hearing Impairment in Affected Individuals From Cameroon (Sub-Saharan Africa)|url=https://www.frontiersin.org/articles/10.3389/fresc.2021.726761/full|journal=Frontiers in Rehabilitation Sciences|volume=2|doi=10.3389/fresc.2021.726761|issn=2673-6861|pmc=9397862|pmid=36188771}}</ref> {{HTitle|Challenges, Opportunities and Notes}} "The Strengthening ear and hearing care capacities in Cameroon" project, implemented by the [https://en.wikipedia.org/wiki/Cameroon%20Baptist%20Convention Cameroon Baptist Convention Health Services] in partnership with the [https://en.wikipedia.org/wiki/CBM_(charity) Christian Blind Mission] and funding from the German Ministry for Economic Development and Cooperation. The aim of the planned project is to improve the quality of life of people with hearing loss or at risk of hearing loss in the following target regions: Northwest, Centre, Littoral. The project will focus on three areas of intervention: Capacity building of existing health professionals through continuing education, access to consulting and treatment services and, development of a national plan to strengthen ear and hearing health care. The aim of the planned project is to improve the quality of life of people with hearing loss or at risk of hearing loss in the following target regions: Northwest, Centre, Littoral. {{HTitle|Audiology Charities}} * The [https://www.bmz.de/en/ German Federal Ministry for Economic Cooperation and Development] (BMZ) * [https://www.cbm.org/ Christian Blind Mission] (CBM) * [https://cbchealthservices.org/ Cameroon Baptist Convention (CBC) Health Services] * [https://terptree.co.uk/communication-professionals/who-are-sound-seekers/ Sound Seekers] (UK) {{HTitle|References}} {{reflist}} {{Global Audiology Authors |name1=Aisha Kinyuy |role1=Author |linkedin1=https://www.linkedin.com/in/aisha-kinyuy-7278471b5/ |name2=Madison Saunders |role2=Contributor |linkedin2=https://www.linkedin.com/in/mcmsaunders/ }} [[Category:Audiology]] [[Category:Cameroon]] h5kfbnwq2r61c6o3akju92vzuz5nwg2 0 328204 2806695 2806647 2026-04-26T19:17:44Z ~2026-25504-03 3068785 /* The Septuagint Chronology */ 2806695 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] 0x0sdt4e62i42czg4jq3py3u0syas4h 2806696 2806695 2026-04-26T19:22:39Z ~2026-25504-03 3068785 /* Septuagint Adjustments */ 2806696 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. ==== The "Whoops Theory" ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] g5ob1ds1udc6z0wggklju2q6eslbqm7 2806697 2806696 2026-04-26T19:28:44Z ~2026-25504-03 3068785 /* Septuagint Adjustments */ 2806697 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ==== The "Whoops Theory" ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] bukumva6c8yrahojivr18tkppd4lhbg 2806702 2806697 2026-04-26T20:00:05Z ~2026-25504-03 3068785 /* The "Whoops Theory" */ 2806702 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ==== The "Whoops Theory": A digression ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="1" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] jmsjhof87d502mwehhds06mg8fl9902 2806703 2806702 2026-04-26T20:01:14Z ~2026-25504-03 3068785 /* The "Whoops Theory": A digression */ 2806703 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ==== The "Whoops Theory": A digression ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="1" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] ek5uwd3ogwknnwyuiuemfznabu25syy 2806704 2806703 2026-04-26T20:02:25Z ~2026-25504-03 3068785 /* The "Whoops Theory": A digression */ 2806704 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ==== The "Whoops Theory": A digression ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] t83yav0bx4dh26q4cgswtgv5ewcvjey 2806705 2806704 2026-04-26T20:05:20Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806705 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ==== The "Whoops Theory": A digression ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] q9franaiovz1h1so0vs1vlyreif4y4h 2806706 2806705 2026-04-26T20:05:52Z ~2026-25504-03 3068785 /* The "Whoops Theory": A digression */ 2806706 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ==== The "Whoops Theory": A digression ==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begat his son, intending to "fix" the timeline, but failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah could suggest the avoidance of a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] m68fj2uvn47ujzxt3e11wptr3lcceat 2806707 2806706 2026-04-26T20:07:56Z ~2026-25504-03 3068785 /* The "Whoops Theory": A digression */ 2806707 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. This interpretation overlooks the systemic nature of the LXX changes. If an editor is methodical enough to systematically alter multiple generations by exactly one hundred years, a single "failure" to fix Methuselah suggests that avoiding a post-Flood death was not the primary objective. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] 7ce9qswelymf5j7dkznk5hvbh3ukecj 2806708 2806707 2026-04-26T20:54:47Z ~2026-25504-03 3068785 /* The "Whoops Theory": A Digression */ 2806708 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood. This presumably places the birth of Shem at 2,164 years—exactly one hundred years before the Flood—aligning his data with the "Long Chronology" of the Septuagint. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] pkdt1te9ds7uv39fvv7qw6yuq5pjai8 2806709 2806708 2026-04-26T21:00:40Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806709 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, which is a much later flood date than is found in other chronologies. In the comment section of the original article, in response to evidence regarding this longer tradition (provided by commenter Roger Quill), Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses to the 187-year begettal age of Methuselah. In this view, Codex Alexandrinus is seen as the lone legitimate witness, while others are discounted: * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. However, the claim that this uncertainty invalidates Demetrius is arguably overstated. As shown in the comparative tables above, there is remarkably little variation in the known traditions: begettal ages almost universally shift by increments of exactly 100 years (e.g., 130 vs. 230). Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the necessary correction for Methuselah—the very figure that would require it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] tw273vvr1czagxa8im78h16jihpjpqy 2806710 2806709 2026-04-26T21:09:50Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806710 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, which is a later flood date than is typically found in other chronologies. In the comment section of the original article, in response to evidence regarding this late flood date, Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses. * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. * '''Codex Alexandrinus:''' is seen as the lone legitimate witness to the The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. A plausible explanation for the discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. However, the claim that this uncertainty invalidates Demetrius is arguably overstated. As shown in the comparative tables above, there is remarkably little variation in the known traditions: begettal ages almost universally shift by increments of exactly 100 years (e.g., 130 vs. 230). Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the necessary correction for Methuselah—the very figure that would require it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] 1zxq0fueaftkdfvak8177g99sypqiyk 2806711 2806710 2026-04-26T21:36:17Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806711 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the comment section of the original article, in response to evidence regarding a longer chronology in which the Methuselah fathering age was '''187''' as opposed to '''167''', Paul D. reaffirms his "Whoops Theory" by challenging the validity of various witnesses. * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 22 years (rather than the typical 20-year variance) whose precise placement remains unknown. * '''Codex Alexandrinus:''' is seen as the lone legitimate witness to the The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, which supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. A plausible explanation for the 22 vs 20 year discrepancy in Demetrius's chronology is the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the necessary correction for Methuselah—the very figure that would require it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] 9yu4pghwiqu8bspbye16b4fs54cnzyz 2806713 2806711 2026-04-26T22:00:28Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806713 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the comment section of the original article, in response to evidence regarding a longer chronology in which the Methuselah fathering age was '''187''' as opposed to '''167''', Paul D. reaffirms his "Whoops Theory" by challenging the validity of various early witnesses. * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 2 years whose precise placement remains unknown. * '''Codex Alexandrinus:''' is seen as the lone legitimate witness to the 187 year fathering age. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, which presumably supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2 year discrepancy in Demetrius's chronology, the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the necessary correction for Methuselah—the very figure that would require it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] 9ucf4ar39wg0oi02s9qjflc06ebesna 2806714 2806713 2026-04-26T22:01:00Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806714 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. In particular, Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the comment section of the original article, in response to evidence regarding a longer chronology in which the Methuselah fathering age was '''187''' as opposed to '''167''', Paul D. reaffirms his "Whoops Theory" by challenging the validity of various early witnesses. * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 2 years whose precise placement remains unknown. * '''Codex Alexandrinus:''' is seen as the lone legitimate witness to the 187 year fathering age. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, which presumably supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2 year discrepancy in Demetrius's chronology, the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the necessary correction for Methuselah—the very figure that would require it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] qvu2wa8qmdoff9daut11tgbpk2uqkpc 2806717 2806714 2026-04-26T22:20:04Z ~2026-25504-03 3068785 /* Septuagint Adjustments */ 2806717 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the comment section of the original article, in response to evidence regarding a longer chronology in which the Methuselah fathering age was '''187''' as opposed to '''167''', Paul D. reaffirms his "Whoops Theory" by challenging the validity of various early witnesses. * '''Josephus:''' Characterized as dependent on the Masoretic tradition. * '''Pseudo-Philo:''' Dismissed due to textual corruption ("a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through the later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional 2 years whose precise placement remains unknown. * '''Codex Alexandrinus:''' is seen as the lone legitimate witness to the 187 year fathering age. The claim that Julius Africanus is invalidated due to his survival through an intermediary, or that Demetrius is disqualified by a 22-year variance, is arguably overstated. Demetrius the Chronographer, writing as early as the late 3rd century BC (c. 221 BC), represents the earliest known witness to biblical chronological calculations. While only fragments of his work remain, they are significant; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, which presumably supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2 year discrepancy in Demetrius's chronology, the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can easily account for such variances without necessitating the rejection of the witnesses. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the necessary correction for Methuselah—the very figure that would require it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] l6rbhxxm2cefvowtudjstrscw2qc0qi 2806718 2806717 2026-04-26T22:29:21Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806718 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the original article's comments, a debate surfaced regarding the longer chronology where Methuselah’s fathering age is recorded as '''187''' rather than '''167'''. Paul D. defends his "Whoops Theory" by systematically challenging the validity of the early witnesses that support the longer timeline: * '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness. * '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as "a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown. * '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age. The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] 888mwm8ezs7zbrjmzwkqfawbss000mm 2806719 2806718 2026-04-26T22:30:12Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806719 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the original article's comments, a debate surfaced regarding the longer chronology where Methuselah’s fathering age is recorded as '''187''' rather than '''167'''. Paul D. defends his "Whoops Theory" by systematically challenging the validity of the early witnesses that support the longer timeline: * '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness. * '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as "a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown. * '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age. The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in patriarchs surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] ibrxlzawqw5jjcwl87e61b4cph2pqgl 2806720 2806719 2026-04-26T22:32:13Z ~2026-25504-03 3068785 /* Demetrius the Chronographer */ 2806720 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the original article's comments, a debate surfaced regarding a longer chronology where Methuselah’s fathering age is recorded as '''187''' rather than '''167'''. Paul D. defends his "Whoops Theory" by systematically challenging the validity of the early witnesses that support the longer timeline: * '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness. * '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as "a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown. * '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age. The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in patriarchs surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all biblical records. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] iar2m6enpzr4m06lwykygrmwd3n77m1 2806723 2806720 2026-04-26T22:37:15Z ~2026-25504-03 3068785 /* Additional Textual Evidence */ 2806723 wikitext text/x-wiki {{Original research}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. == Summary of Main Arguments == The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include: * '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or "perfect" numbers (such as 7, 49, and 60) rather than biological or historical reality. * '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions. * '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood. * '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific "years of the world" (''Anno Mundi''), synchronizing human history with a divine calendar. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . . *During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar. . . . This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. ==== Mesopotamian Similarities ==== *[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions. *[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. ==== Demetrius the Chronographer ==== In the original article's comments, a debate surfaced regarding a longer chronology where Methuselah’s fathering age is recorded as '''187''' rather than '''167'''. Paul D. defends his "Whoops Theory" by systematically challenging the validity of the early witnesses that support the longer timeline: * '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness. * '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as "a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown. * '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age. The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports the '''187''' fathering age. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. The reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable. '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in patriarchs surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} <small>* '''Dash (—)''' indicates where primary sources do not provide complete death data.</small> * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. == The Birth of Shem (A Digression) == Were Noah's sons born when Noah was 500 or 502? ==== The 502 Calculation ==== While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block">\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209</math> ** Post-Flood Patriarch years: *:<math display="block">\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241</math> ** Total Years: *:<math display="block">7^4 + 7^2 = 2401 + 49 = 2450</math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block">7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}</math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block">\begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } & 70 \times 40 = 2800 \end{aligned}</math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. = End TBD = '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that could result in patriarchs surviving beyond the date of the Flood. * <span style="color:#333333;">'''Blank Cells'''</span> indicate where primary sources do not provide specific lifespan or death data. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == The Septuagint Chronology == === The Correlations === An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo). The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the "Universal Flood" problem. [[Category:Religion]] q3g285zxrofbxsuc1jdxued4dgsqs32 0 328786 2806753 2805379 2026-04-27T04:19:59Z OhanaUnited 18921 typo 2806753 wikitext text/x-wiki {{Article info | first1 = Andrii | last1 = Boida | orcid1 = 0000-0001-7062-1978 | affiliation1 = Vasyl Stefanyk Carpathian National University | submitted = 2026-03-20 | correspondence1 = {{nospam|boida29|serpnia.gmail.com}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The article analyzes the author’s contribution to the substantial improvement of the visual presentation and informational content of the Wikipedia article about the People’s Commissar of Justice and Prosecutor General of the Ukrainian SSR (1927–1930), as well as the commander of the Red Ukrainian Galician Army (1920). The assistance of other contributors, who managed to discover sources previously unknown to the author and helped elevate the article to “Good Article” status, is evaluated. The prospects for studying the figure of Vasyl Porayko through the medium of Wikipedia are assessed, and further forecasts are made regarding the article’s potential promotion to “Featured Article” status. Special emphasis is placed on how the author’s experience working on [https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B9%D0%BA%D0%BE_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C_%D0%86%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87 Vasyl Porayko]’s biography contributes to the writing of dissertation on this topic. ''Keywords:'' Vasyl Porayko, Galicia, First World War, Ukrainian SSR, Wikipedia. У статті аналізується вклад автора у суттєве покращення зовнішнього виду та інформаційного наповнення статті у Вікіпедії про наркома юстиції та генпрокурора УСРР (1927 – 1930), а також командарма Червоної Української Галицької армії (1920). Дається оцінка допомоги інших користувачів, які змогли віднайти раніше невідомі автору джерела і вивести статтю на статус доброї. Оцінюються перспективи вивчення постаті Василя Порайка крізь простір Вікіпедії та даються подальші прогнози щодо виведення статті у статус вибраних. Окремий акцент ставиться на користь досвіду роботи над біографією [https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B9%D0%BA%D0%BE_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C_%D0%86%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87 Василя Порайка] щодо написання автором дисертації на цю тему. ''Ключові слова:'' Василь Порайко, Галичина, Перша світова війна, УРСР, Вікіпедія. pzxzczgx8y89whg7cfqqy3y6klvpgku 2806754 2806753 2026-04-27T04:22:08Z OhanaUnited 18921 bypass 2806754 wikitext text/x-wiki {{Article info | first1 = Andrii | last1 = Boida | orcid1 = 0000-0001-7062-1978 | affiliation1 = Vasyl Stefanyk Carpathian National University | submitted = | correspondence1 = {{nospam|boida29|serpnia.gmail.com}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The article analyzes the author’s contribution to the substantial improvement of the visual presentation and informational content of the Wikipedia article about the People’s Commissar of Justice and Prosecutor General of the Ukrainian SSR (1927–1930), as well as the commander of the Red Ukrainian Galician Army (1920). The assistance of other contributors, who managed to discover sources previously unknown to the author and helped elevate the article to “Good Article” status, is evaluated. The prospects for studying the figure of Vasyl Porayko through the medium of Wikipedia are assessed, and further forecasts are made regarding the article’s potential promotion to “Featured Article” status. Special emphasis is placed on how the author’s experience working on [https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B9%D0%BA%D0%BE_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C_%D0%86%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87 Vasyl Porayko]’s biography contributes to the writing of dissertation on this topic. ''Keywords:'' Vasyl Porayko, Galicia, First World War, Ukrainian SSR, Wikipedia. У статті аналізується вклад автора у суттєве покращення зовнішнього виду та інформаційного наповнення статті у Вікіпедії про наркома юстиції та генпрокурора УСРР (1927 – 1930), а також командарма Червоної Української Галицької армії (1920). Дається оцінка допомоги інших користувачів, які змогли віднайти раніше невідомі автору джерела і вивести статтю на статус доброї. Оцінюються перспективи вивчення постаті Василя Порайка крізь простір Вікіпедії та даються подальші прогнози щодо виведення статті у статус вибраних. Окремий акцент ставиться на користь досвіду роботи над біографією [https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B9%D0%BA%D0%BE_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C_%D0%86%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87 Василя Порайка] щодо написання автором дисертації на цю тему. ''Ключові слова:'' Василь Порайко, Галичина, Перша світова війна, УРСР, Вікіпедія. h9vfd7r951btjbephznx527uzi7ruxd 2806755 2806754 2026-04-27T04:22:27Z OhanaUnited 18921 + 2806755 wikitext text/x-wiki {{Article info | first1 = Andrii | last1 = Boida | orcid1 = 0000-0001-7062-1978 | affiliation1 = Vasyl Stefanyk Carpathian National University | submitted = 24 Jun 2025 | correspondence1 = {{nospam|boida29|serpnia.gmail.com}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The article analyzes the author’s contribution to the substantial improvement of the visual presentation and informational content of the Wikipedia article about the People’s Commissar of Justice and Prosecutor General of the Ukrainian SSR (1927–1930), as well as the commander of the Red Ukrainian Galician Army (1920). The assistance of other contributors, who managed to discover sources previously unknown to the author and helped elevate the article to “Good Article” status, is evaluated. The prospects for studying the figure of Vasyl Porayko through the medium of Wikipedia are assessed, and further forecasts are made regarding the article’s potential promotion to “Featured Article” status. Special emphasis is placed on how the author’s experience working on [https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B9%D0%BA%D0%BE_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C_%D0%86%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87 Vasyl Porayko]’s biography contributes to the writing of dissertation on this topic. ''Keywords:'' Vasyl Porayko, Galicia, First World War, Ukrainian SSR, Wikipedia. У статті аналізується вклад автора у суттєве покращення зовнішнього виду та інформаційного наповнення статті у Вікіпедії про наркома юстиції та генпрокурора УСРР (1927 – 1930), а також командарма Червоної Української Галицької армії (1920). Дається оцінка допомоги інших користувачів, які змогли віднайти раніше невідомі автору джерела і вивести статтю на статус доброї. Оцінюються перспективи вивчення постаті Василя Порайка крізь простір Вікіпедії та даються подальші прогнози щодо виведення статті у статус вибраних. Окремий акцент ставиться на користь досвіду роботи над біографією [https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D0%B0%D0%B9%D0%BA%D0%BE_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C_%D0%86%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87 Василя Порайка] щодо написання автором дисертації на цю тему. ''Ключові слова:'' Василь Порайко, Галичина, Перша світова війна, УРСР, Вікіпедія. ifs8r97u10yxk8wyxh4h0y9of08l5c1 0 328807 2806745 2805988 2026-04-27T04:10:40Z OhanaUnited 18921 fix 2806745 wikitext text/x-wiki {{Article info | first1 = Halyna | last1 = Kachurovska | orcid1 = | affiliation1 = | submitted = 2026-03-20 | correspondence1 = {{nospam|lotos27|meta.ua serpnia.gmail.com}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The article analyzes the coverage of the life, work and cultural heritage of the Polish artist Jan Matejko in the Ukrainian Wikipedia and its sister projects. The features of the article structure[TM4.1], the representation of the artist’s works and related historical figures and locations in Ukraine are considered. Special attention is paid to the Ukrainian context of the artist’s legacy, his influence on the development of art in Galicia, and the works of his students. The importance of Wikipedia as an important tool for research, popularization and preservation of cultural heritage is emphasized. ''Keywords:'' Jan Matejko, cultural heritage, Wikipedia, Wikimedia Commons, Wikidata, Ukrainian-Polish cultural ties. У статті проаналізовано висвітлення життя, творчості та культурної спадщини польського художника Яна Матейка в українській Вікіпедії та її сестринських проєктах. Розглядаються особливості структури статей, представленість творів митця та пов’язаних із ним історичних постатей і локацій в Україні. Окрему увагу приділено українському контексту спадщини художника, його впливу на розвиток мистецтва в Галичині, роботам його учнів. Підкреслено значення Вікіпедії як важливого інструменту для дослідження, популяризації та збереження культурної спадщини. ''Ключові слова:'' Ян Матейко, культурна спадщина, Вікіпедія, Вікісховище, Вікідані, українсько-польські культурні зв’язки. ==== Articles created (in chronological order) in the Ukrainian Wikipedia related to the life and work of Jan Matejko''.'' ==== ===== 2023: ===== * [https://uk.wikipedia.org/wiki/Площа_Яна_Матейка_(Краків) Площа Яна Матейка] (Краків) * [https://uk.wikipedia.org/wiki/Анджей_Беднарчик Анджей Беднарчик] (ректор Академії ОМ ім. Я. Матейка ) * [https://uk.wikipedia.org/wiki/Брама_Флоріанська_%28Краків%29 Брама Флоріанська] (Краків) (проєкт Я. Матейка) '''2025:''' * [https://uk.wikipedia.org/wiki/Гелена_Унєжиська Гелена Унєжиська] (донька) * [https://uk.wikipedia.org/wiki/Юзеф_Унєжиський Юзеф Унєжиський] (учень і чоловік Гелени ) * [https://uk.wikipedia.org/wiki/Марія_Голіховська Марія Голіховська] (сестра Матейка) * [https://uk.wikipedia.org/wiki/Пам%27ятник_Яну_Матейку_%28Краків%29 Пам'ятник Яну Матейку] (Краків) * [https://uk.wikipedia.org/wiki/Пам%27ятник_Яну_Матейку_%28Варшава%29 Пам'ятник Яну Матейку] (Варшава) '''2026:''' * [https://uk.wikipedia.org/wiki/Пауліна_Ґібултовська Пауліна Ґібултовська] (мати дружини) * [https://uk.wikipedia.org/wiki/Ольга_Фіалка Ольга Фіалка] (учениця) '''Categories and sub-categories created:''' * Категорія:Ян Матейко * Категорія:Учні Яна Матейка * Категорія:Родина Яна Матейка * Kатегорія:Пам'ятники Яну Матейку j9fsn4rmdc5p2hbygmnh4wfit14owzw 2806757 2806745 2026-04-27T04:23:48Z OhanaUnited 18921 submit date 2806757 wikitext text/x-wiki {{Article info | first1 = Halyna | last1 = Kachurovska | orcid1 = | affiliation1 = | submitted = 24 Jun 2025 | correspondence1 = {{nospam|lotos27|meta.ua serpnia.gmail.com}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The article analyzes the coverage of the life, work and cultural heritage of the Polish artist Jan Matejko in the Ukrainian Wikipedia and its sister projects. The features of the article structure[TM4.1], the representation of the artist’s works and related historical figures and locations in Ukraine are considered. Special attention is paid to the Ukrainian context of the artist’s legacy, his influence on the development of art in Galicia, and the works of his students. The importance of Wikipedia as an important tool for research, popularization and preservation of cultural heritage is emphasized. ''Keywords:'' Jan Matejko, cultural heritage, Wikipedia, Wikimedia Commons, Wikidata, Ukrainian-Polish cultural ties. У статті проаналізовано висвітлення життя, творчості та культурної спадщини польського художника Яна Матейка в українській Вікіпедії та її сестринських проєктах. Розглядаються особливості структури статей, представленість творів митця та пов’язаних із ним історичних постатей і локацій в Україні. Окрему увагу приділено українському контексту спадщини художника, його впливу на розвиток мистецтва в Галичині, роботам його учнів. Підкреслено значення Вікіпедії як важливого інструменту для дослідження, популяризації та збереження культурної спадщини. ''Ключові слова:'' Ян Матейко, культурна спадщина, Вікіпедія, Вікісховище, Вікідані, українсько-польські культурні зв’язки. ==== Articles created (in chronological order) in the Ukrainian Wikipedia related to the life and work of Jan Matejko''.'' ==== ===== 2023: ===== * [https://uk.wikipedia.org/wiki/Площа_Яна_Матейка_(Краків) Площа Яна Матейка] (Краків) * [https://uk.wikipedia.org/wiki/Анджей_Беднарчик Анджей Беднарчик] (ректор Академії ОМ ім. Я. Матейка ) * [https://uk.wikipedia.org/wiki/Брама_Флоріанська_%28Краків%29 Брама Флоріанська] (Краків) (проєкт Я. Матейка) '''2025:''' * [https://uk.wikipedia.org/wiki/Гелена_Унєжиська Гелена Унєжиська] (донька) * [https://uk.wikipedia.org/wiki/Юзеф_Унєжиський Юзеф Унєжиський] (учень і чоловік Гелени ) * [https://uk.wikipedia.org/wiki/Марія_Голіховська Марія Голіховська] (сестра Матейка) * [https://uk.wikipedia.org/wiki/Пам%27ятник_Яну_Матейку_%28Краків%29 Пам'ятник Яну Матейку] (Краків) * [https://uk.wikipedia.org/wiki/Пам%27ятник_Яну_Матейку_%28Варшава%29 Пам'ятник Яну Матейку] (Варшава) '''2026:''' * [https://uk.wikipedia.org/wiki/Пауліна_Ґібултовська Пауліна Ґібултовська] (мати дружини) * [https://uk.wikipedia.org/wiki/Ольга_Фіалка Ольга Фіалка] (учениця) '''Categories and sub-categories created:''' * Категорія:Ян Матейко * Категорія:Учні Яна Матейка * Категорія:Родина Яна Матейка * Kатегорія:Пам'ятники Яну Матейку 8l3858zm1cdaw8yw864mszahne9w8q8 2806763 2806757 2026-04-27T04:37:04Z OhanaUnited 18921 verified with conference organizer for correct email 2806763 wikitext text/x-wiki {{Article info | first1 = Halyna | last1 = Kachurovska | orcid1 = | affiliation1 = | submitted = 24 Jun 2025 | correspondence1 = {{nospam|lotos27|meta.ua}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The article analyzes the coverage of the life, work and cultural heritage of the Polish artist Jan Matejko in the Ukrainian Wikipedia and its sister projects. The features of the article structure[TM4.1], the representation of the artist’s works and related historical figures and locations in Ukraine are considered. Special attention is paid to the Ukrainian context of the artist’s legacy, his influence on the development of art in Galicia, and the works of his students. The importance of Wikipedia as an important tool for research, popularization and preservation of cultural heritage is emphasized. ''Keywords:'' Jan Matejko, cultural heritage, Wikipedia, Wikimedia Commons, Wikidata, Ukrainian-Polish cultural ties. У статті проаналізовано висвітлення життя, творчості та культурної спадщини польського художника Яна Матейка в українській Вікіпедії та її сестринських проєктах. Розглядаються особливості структури статей, представленість творів митця та пов’язаних із ним історичних постатей і локацій в Україні. Окрему увагу приділено українському контексту спадщини художника, його впливу на розвиток мистецтва в Галичині, роботам його учнів. Підкреслено значення Вікіпедії як важливого інструменту для дослідження, популяризації та збереження культурної спадщини. ''Ключові слова:'' Ян Матейко, культурна спадщина, Вікіпедія, Вікісховище, Вікідані, українсько-польські культурні зв’язки. ==== Articles created (in chronological order) in the Ukrainian Wikipedia related to the life and work of Jan Matejko''.'' ==== ===== 2023: ===== * [https://uk.wikipedia.org/wiki/Площа_Яна_Матейка_(Краків) Площа Яна Матейка] (Краків) * [https://uk.wikipedia.org/wiki/Анджей_Беднарчик Анджей Беднарчик] (ректор Академії ОМ ім. Я. Матейка ) * [https://uk.wikipedia.org/wiki/Брама_Флоріанська_%28Краків%29 Брама Флоріанська] (Краків) (проєкт Я. Матейка) '''2025:''' * [https://uk.wikipedia.org/wiki/Гелена_Унєжиська Гелена Унєжиська] (донька) * [https://uk.wikipedia.org/wiki/Юзеф_Унєжиський Юзеф Унєжиський] (учень і чоловік Гелени ) * [https://uk.wikipedia.org/wiki/Марія_Голіховська Марія Голіховська] (сестра Матейка) * [https://uk.wikipedia.org/wiki/Пам%27ятник_Яну_Матейку_%28Краків%29 Пам'ятник Яну Матейку] (Краків) * [https://uk.wikipedia.org/wiki/Пам%27ятник_Яну_Матейку_%28Варшава%29 Пам'ятник Яну Матейку] (Варшава) '''2026:''' * [https://uk.wikipedia.org/wiki/Пауліна_Ґібултовська Пауліна Ґібултовська] (мати дружини) * [https://uk.wikipedia.org/wiki/Ольга_Фіалка Ольга Фіалка] (учениця) '''Categories and sub-categories created:''' * Категорія:Ян Матейко * Категорія:Учні Яна Матейка * Категорія:Родина Яна Матейка * Kатегорія:Пам'ятники Яну Матейку r4nxiplod8iwsckzlh8887yiyqsn544 0 328809 2806759 2806017 2026-04-27T04:27:42Z OhanaUnited 18921 missing Ukrainian key words 2806759 wikitext text/x-wiki {{Article info | first1 = Yuri | last1 = Perohanych | orcid1 = 0000-0003-0140-3146 | affiliation1 = Association of Information Technology Enterprises of Ukraine | correspondence1 = {{nospam|perohanych|gmail.com}} | w1 = | journal = WikiJournal of Humanities | license = | abstract = | submitted = 2025-06-24 }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' This article examines the development of regional wiki projects in Ukraine as an important component of contemporary digital humanities. Local electronic encyclopedias created with wiki technologies play a key role in collecting, preserving, and disseminating the historical and cultural heritage of communities. The study analyses the structure and practices of several notable projects, including Encyclopedia of Nosivshchyna, Ternopedia, Akkermanika, and WikiBoyarka, which represent different models of organization, collaboration, and content development. The advantages of wiki platforms – openness, dynamism, and integration into the global information ecosystem – are highlighted, along with the key challenges such as ensuring content verifiability, standardization of article structure, and expanding contributor communities. The article emphasizes the significance of regional wiki projects for strengthening local identity and safeguarding cultural memory in the digital age. ''Keywords:'' wikiprojects, local history, local electronic encyclopedias, cultural heritage, digital humanities. У статті проаналізовано розвиток краєзнавчих вікіпроєктів в Україні як важливого сегмента сучасної цифрової гуманітаристики. Показано, що локальні електронні енциклопедії, створені на основі вікітехнологій, відіграють ключову роль у збиранні, збереженні та популяризації історико-культурної спадщини територіальних громад. Розглянуто особливості та досвід таких проєктів, як «Енциклопедія Носівщини», «Тернопедія», «Аккерманіка» та «ВікіБоярка», які демонструють різні моделі організації, співпраці та наповнення контенту. Визначено основні переваги вікіплатформ у краєзнавстві – відкритість, динамічність, інтегрованість у глобальний інформаційний простір – а також окреслено ключові виклики, серед яких необхідність уніфікації джерельної бази, забезпечення якості матеріалів та залучення нових учасників. Підкреслено значення краєзнавчих вікіпроєктів для формування локальної ідентичності та збереження культурної пам’яті в умовах цифрової доби. ''Ключові слова:'' вікіпроєкти, краєзнавство, локальні електронні енциклопедії, культурна спадщина, цифрова гуманітаристика. ''Ключові слова:'' вікіпроєкти, краєзнавство, локальні електронні енциклопедії, культурна спадщина, цифрова гуманітаристика. dovqmf6uplolfu81m4z5ps2h2yf69f4 3 328990 2806684 2806633 2026-04-26T16:53:31Z Juandev 2651 /* Do not remove content */ new section 2806684 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Dronebogus!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. 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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:36, 10 April 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} ==Reversion== FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FCortical_structures_and_motivational_drive&diff=2803999&oldid=2763251 this edit]. Feel free to discuss. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:38, 10 April 2026 (UTC) FYI, I've reinstated [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2023%2FFlourishing_in_the_elderly&diff=2803279&oldid=2798973 this removal of a figure]. Please discuss if you think an image shouldn't be used. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:42, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FLucid_dream_facilitation&diff=2801417&oldid=2677316 this image removal]. Please discuss before removal. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:58, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FTattoo_regret&diff=2798977&oldid=2761836 this image removal]. Please discuss before removal. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:20, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSpirituality_and_mental_health&diff=2798806&oldid=2758870 this image removal]. Please engage in constructive editng by discussing if you have concerns about an image and/or providing a better alternative image. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:31, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FSleep_and_ego_depletion&diff=2798553&oldid=2758863 this image removal]. Please discuss concerns or suggest a suitable alternative. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:30, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2024%2FAyahuasca_and_the_brain&diff=2796791&oldid=2688733 this image removal]. Discuss and/or provide alternative. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:35, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FBoredom_and_substance_use&diff=2795132&oldid=2758605 this image removal]]. No edit summary. Discuss and/or provide alternative. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:51, 10 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Lectures/Brain_and_physiological_needs&diff=prev&oldid=2806621 this image change]. No edit summary. Feel free to discuss. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:23, 26 April 2026 (UTC) FYI, I've reverted [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FStockholm_syndrome_emotion&diff=2806623&oldid=2806515 this image change]. It is unclear how the proposed image relates to the topic. Feel free to discuss. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:36, 26 April 2026 (UTC) ==Disruptive editing warning== This is a first warning for disruptive editing (edit warring, not engaging in constructive dialogue). You are out of your lane. Continue and you will be blocked on this project. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:06, 10 April 2026 (UTC) == Do not remove content == Dear Dronebogus, FYI AI generated images [[Wikiversity:Artificial intelligence|are allowed on Wikiversity]]. So stop replacing them by other media if the authors of the resource doesnt agree. Thanks. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:53, 26 April 2026 (UTC) svt69a7nd4br9z644uqfjl4tc86ehh9 1 329290 2806742 2806668 2026-04-27T04:04:34Z Dronebogus 3054149 /* AI slop */ Reply 2806742 wikitext text/x-wiki == AI slop == {{ping|Jtneill}} yet another case of using AI slop to represent something that cannot realistically be illustrated even with real images. Empathy, just like schadenfreude or disappointment, is contextual— it’s defined entirely by one’s reaction to another’s emotions. There is no empathy in isolation, and no facial expression that can independently convey an emotion that exists across a spectrum. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 06:00, 26 April 2026 (UTC) :There is lot of ongoing work in understanding the facial expression of emotion. Emotion recognition from facial expressions is a trainable part of emotional intelligence. For example, you may be interested in the [[w:Facial Action Coding System]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:59, 26 April 2026 (UTC) ::I don’t see what that has to do with AI slop [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 11:02, 26 April 2026 (UTC) :::I think the image is AI useful. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:05, 26 April 2026 (UTC) ::::That isn’t an AInswer [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:04, 27 April 2026 (UTC) 2q9m8rj58dsdk5xa300dv95idehgiwr 2 329291 2806678 2806658 2026-04-26T14:19:58Z Atcovi 276019 Atcovi moved page [[Tetrad Formation]] to [[User:Notmyfridaybest/Tetrad Formation]] without leaving a redirect: moving under userspace 2806658 wikitext text/x-wiki Tetrad formation is the stage in pollen development where a single diploid pollen mother cell (PMC) undergoes meiosis and produces a group of four haploid microspores arranged together. This cluster of four microspores is called a microspore tetrad. In the anther’s microsporangium, each PMC divides meiotically to form four daughter cells, each with half the chromosome number (n). These four microspores remain temporarily joined in a tetrad, and their arrangement (linear, tetrahedral, etc.) depends on the pattern of cytokinesis (successive or simultaneous). As the anther matures and dehydrates, the microspores separate from the tetrad and each develops into an individual pollen grain through micro gametogenesis. 6uxhfnrzs05nlkqibaf589ddq95jk4i 0 329294 2806679 2026-04-26T14:56:06Z Soboyed 3063058 Created page with "{{:DesignWriteStudio/SiteElements/Navbox}} == Assignment 4.2: Midsemester Portfolio == === Introduction === This portfolio analyzes my transcript archive from the DesignWriteStudio course, covering nine conversations between April 2 and April 25, 2026. I chose to limit the archive to course-related transcripts only, as my AI use this semester has been almost entirely academic. The archive spans assignments 1.1 through 3.3 and was analyzed using Gemini to produce three..." 2806679 wikitext text/x-wiki {{:DesignWriteStudio/SiteElements/Navbox}} == Assignment 4.2: Midsemester Portfolio == === Introduction === This portfolio analyzes my transcript archive from the DesignWriteStudio course, covering nine conversations between April 2 and April 25, 2026. I chose to limit the archive to course-related transcripts only, as my AI use this semester has been almost entirely academic. The archive spans assignments 1.1 through 3.3 and was analyzed using Gemini to produce three distinct readings of the same data: sequential, categorical, and behavioral. === Sequential Overview === The archive contains nine separate conversations spanning April 2 to April 25, 2026. The first prompt was a tightly constrained request to define hypertext using only five specific ACM papers. The most recent was a prompt connecting prior feature analysis to broader design challenges using past transcripts as evidence. Two distinct clusters emerged. The first ran from April 2–4, covering early conceptual work: defining hypertext, establishing the six core features framework, and drafting the first encyclopedia entries. The second was a single intensive afternoon on April 25 where four complex analytical tasks were completed back to back — Façade, Discogs, and the design challenges entries. Between these clusters were two significant gaps: eight days between April 4 and April 12, and thirteen days between April 12 and April 25. The gaps reflect the assignment schedule more than disengagement — work happened in concentrated bursts when deadlines landed. === Skills Overview === Gemini identified four categories of prompting behavior: '''Constructive Output (~40%)''' — The dominant mode. I provide raw materials (PDFs, source files, prior entries) and ask for a structured final product with specific word counts, syntax requirements, and citation rules. '''Recursive Contextualization (~30%)''' — I frequently "prime" sessions by feeding the model its own previous outputs or course materials to maintain continuity across conversations. This is what the assignment calls "seeding the model." '''Critical Audit & Validation (~20%)''' — Rather than asking for general feedback, I ask the model to check outputs against external standards — Wikipedia's verifiability rules, DOI accuracy, and the distinction between sourced claims and original synthesis. '''Analytical Application (~10%)''' — Applying the six core features framework to a specific case study like Façade or Discogs. Behaviors I never use: open-ended brainstorming, general information seeking, debugging or coding, emotional or social chatting, and vague task delegation. Every prompt is bounded by constraints, source files, and formatting requirements. === Operator Assessment === Gemini's assessment was blunt: I operate as an Architect and Auditor. I do not chat with the model — I manage it. The defining traits are high-control redirection, systematic pushback, iterative follow-ups, and zero tolerance for hallucination. I never let the model lead. Every session begins with a priming prompt that strips the AI of its general knowledge and forces it to operate within the course framework. When the model drifts — hallucinating a connection between Façade and the software "ink," or citing the wrong DOI — I identify and correct it explicitly. I treat AI output as a draft to be audited, not a finished answer to be accepted. My workflow follows a consistent loop: Feed Data → Request Draft → Audit for Errors → Finalize. The output of one task becomes the required context for the next. === What Surprised Me === The "Recursive Contextualization" category surprised me most. I hadn't consciously thought of the priming prompts as a behavior pattern — I was just doing what the assignment required. But seeing it quantified at 30% of my prompts made me realize how much of my effort goes into compensating for the model's lack of persistent memory. I'm not just using the AI to produce content; I'm spending nearly a third of my prompts rebuilding its context from scratch every session. That's a real cost that isn't visible when you're inside the workflow. {{:DesignWriteStudio/SiteElements/Footer}} k8pskrn0nni6z1cy994w3jfo0bieony 0 329295 2806680 2026-04-26T15:18:08Z Soboyed 3063058 Created page with "{{:DesignWriteStudio/SiteElements/Navbox}} == 5.1 Data as Hypertextual Objects: Africa == === Overview === This page explores the representation of African sovereign states through structured data retrieved from Wikidata. By organizing information such as population and land area into hypertextual tables, we can examine how different metrics shape our perception of a continent's geopolitical landscape. The following data was retrieved via SPARQL queries focusing on ite..." 2806680 wikitext text/x-wiki {{:DesignWriteStudio/SiteElements/Navbox}} == 5.1 Data as Hypertextual Objects: Africa == === Overview === This page explores the representation of African sovereign states through structured data retrieved from Wikidata. By organizing information such as population and land area into hypertextual tables, we can examine how different metrics shape our perception of a continent's geopolitical landscape. The following data was retrieved via SPARQL queries focusing on items classified as sovereign states (P31: Q3624078) located in Africa (P30: Q15). === All African States === {| class="wikitable sortable" !Country !Population !Area (km²) |- |Algeria |46,164,219 |2,381,741 |- |Angola |36,749,906 |1,246,700 |- |Benin |14,111,034 |114,763 |- |Botswana |2,480,244 |581,737 |- |Burkina Faso |23,025,776 |274,200 |- |Burundi |13,689,450 |27,834 |- |Cameroon |28,372,687 |475,442 |- |Cape Verde |555,988 |4,033 |- |Central African Republic |5,152,421 |622,984 |- |Chad |19,319,064 |1,284,000 |- |Comoros |902,348 |2,034 |- |Democratic Republic of the Congo |105,789,731 |2,344,858 |- |Djibouti |1,152,944 |23,200 |- |Egypt |114,535,772 |1,010,407 |- |Equatorial Guinea |1,847,549 |28,051 |- |Eritrea |3,497,000 |117,600 |- |Eswatini |1,230,506 |17,364 |- |Ethiopia |128,691,692 |1,104,300 |- |Gabon |2,025,137 |267,667 |- |Ghana |32,833,031 |238,535 |- |Guinea |12,717,176 |245,857 |- |Guinea-Bissau |1,861,283 |36,125 |- |Ivory Coast |31,165,654 |322,463 |- |Kenya |48,468,138 |581,309 |- |Lesotho |2,007,201 |30,355 |- |Liberia |5,214,030 |111,369 |- |Libya |7,381,023 |1,759,541 |- |Madagascar |31,964,956 |587,295 |- |Malawi |17,563,749 |118,484 |- |Mali |20,250,833 |1,240,192 |- |Mauritania |4,736,139 |1,030,700 |- |Mauritius |1,264,613 |2,040 |- |Morocco |36,828,330 |446,550 |- |Mozambique |29,668,834 |801,590 |- |Namibia |2,533,794 |825,615 |- |Niger |21,477,348 |1,267,000 |- |Nigeria |211,400,708 |923,768 |- |Republic of the Congo |6,142,180 |342,000 |- |Rwanda |14,569,341 |26,338 |- |Senegal |16,876,720 |196,722 |- |Seychelles |119,773 |459 |- |Sierra Leone |7,557,212 |71,740 |- |Somalia |11,031,386 |637,657 |- |South Africa |62,027,503 |1,221,037 |- |South Sudan |12,575,714 |644,329 |- |Sudan |40,533,330 |1,886,068 |- |São Tomé and Príncipe |204,327 |1,001 |- |Tanzania |57,310,019 |947,303 |- |The Gambia |2,639,916 |11,300 |- |Togo |7,797,694 |56,785 |- |Tunisia |11,565,204 |163,610 |- |Uganda |47,123,531 |241,038 |- |Zambia |19,610,769 |752,618 |- |Zimbabwe |15,178,979 |390,757 |} === Top 5 by Population === {| class="wikitable sortable" !Rank !Country !Population |- |1 |Nigeria |211,400,708 |- |2 |Ethiopia |128,691,692 |- |3 |Egypt |114,535,772 |- |4 |Democratic Republic of the Congo |105,789,731 |- |5 |South Africa |62,027,503 |} <ref>Data retrieved from query.wikidata.org for Africa (Q15) on April 26, 2026.</ref> === Top 5 by Area === {| class="wikitable sortable" !Rank !Country !Area (km²) |- |1 |Algeria |2,381,741 |- |2 |Democratic Republic of the Congo |2,344,858 |- |3 |Sudan |1,886,068 |- |4 |Libya |1,759,541 |- |5 |Chad |1,284,000 |} <ref>Data retrieved from query.wikidata.org for Africa (Q15) on April 26, 2026.</ref> === Analysis === The two top-5 lists present distinct narratives regarding regional influence and national importance. Sorting by population highlights Nigeria as the dominant demographic force on the continent, centering the argument about Africa on human capital, labor markets, and urban density. In contrast, sorting by area emphasizes the territorial vastness of Northern and Central African states like Algeria and Chad, framing the continent's importance through land resources and geographical scale. Neither list is neutral; a population-centric view may overlook the logistical and resource-management challenges faced by sprawling, less-populated nations, while an area-centric view ignores the dynamic socio-political influence of high-density population centers. === Data Quality === The raw dataset contains notable anomalies that reflect the complexity of open-source linked data: * '''Duplicate Entries:''' Sudan appears twice in the area dataset with slightly different values (1,886,068 km² and 1,840,687 km²), likely due to different administrative definitions or historical border changes. * '''Missing Data:''' Not all sovereign states may appear in the combined query if either the population or area property was missing for a specific Wikidata item at the time of the query. * '''Precision Variation:''' The area for Egypt is provided with high decimal precision (1,010,407.87), whereas most other countries are rounded to the nearest whole number, indicating inconsistent data entry sources. == References == <references /> {{:DesignWriteStudio/SiteElements/Footer}} 4i4z2roswxoihm51btmrpxtwbxt6jij 0 329296 2806682 2026-04-26T15:45:20Z Soboyed 3063058 Created page with "{{:DesignWriteStudio/SiteElements/Navbox}} == 5.2 Hypertextualizing Data: Africa == === Source Data (Transcluded) === {{:DesignWriteStudio/Course/StudentPages/David/5.1 Data as Hypertextual Objects}} === All African States === <div class="toccolours mw-collapsible mw-collapsed" style="width: 100%; overflow: auto;"><div style="font-weight: bold; padding: 5px;">Click "Expand" to view the full list of African States</div><div class="mw-collapsible-content"> {| class="wik..." 2806682 wikitext text/x-wiki {{:DesignWriteStudio/SiteElements/Navbox}} == 5.2 Hypertextualizing Data: Africa == === Source Data (Transcluded) === {{:DesignWriteStudio/Course/StudentPages/David/5.1 Data as Hypertextual Objects}} === All African States === <div class="toccolours mw-collapsible mw-collapsed" style="width: 100%; overflow: auto;"><div style="font-weight: bold; padding: 5px;">Click "Expand" to view the full list of African States</div><div class="mw-collapsible-content"> {| class="wikitable sortable" style="width: 100%;" !Country !Population !Area (km²) |- |[[Algeria]] |46,164,219 |2,381,741 |- |[[Angola]] |36,749,906 |1,246,700 |- |[[Benin]] |14,111,034 |114,763 |- |[[Botswana]] |2,480,244 |581,737 |- |[[Burkina Faso]] |23,025,776 |274,200 |- |[[Burundi]] |13,689,450 |27,834 |- |[[Cameroon]] |28,372,687 |475,442 |- |[[Cabo Verde|Cape Verde]] |555,988 |4,033 |- |[[Central African Republic]] |5,152,421 |622,984 |- |[[Chad]] |19,319,064 |1,284,000 |- |[[Comoros]] |902,348 |2,034 |- |[[Democratic Republic of the Congo]] |105,789,731 |2,344,858 |- |[[Djibouti]] |1,152,944 |23,200 |- |[[Egypt]] |114,535,772 |1,010,407 |- |[[Equatorial Guinea]] |1,847,549 |28,051 |- |[[Eritrea]] |3,497,000 |117,600 |- |[[Eswatini]] |1,230,506 |17,364 |- |[[Ethiopia]] |128,691,692 |1,104,300 |- |[[Gabon]] |2,025,137 |267,667 |- |[[Ghana]] |32,833,031 |238,535 |- |[[Guinea]] |12,717,176 |245,857 |- |[[Guinea-Bissau]] |1,861,283 |36,125 |- |[[Côte d'Ivoire|Ivory Coast]] |31,165,654 |322,463 |- |[[Kenya]] |48,468,138 |581,309 |- |[[Lesotho]] |2,007,201 |30,355 |- |[[Liberia]] |5,214,030 |111,369 |- |[[Libya]] |7,381,023 |1,759,541 |- |[[Madagascar]] |31,964,956 |587,295 |- |[[Malawi]] |17,563,749 |118,484 |- |[[Mali]] |20,250,833 |1,240,192 |- |[[Mauritania]] |4,736,139 |1,030,700 |- |[[Mauritius]] |1,264,613 |2,040 |- |[[Morocco]] |36,828,330 |446,550 |- |[[Mozambique]] |29,668,834 |801,590 |- |[[Namibia]] |2,533,794 |825,615 |- |[[Niger]] |21,477,348 |1,267,000 |- |[[Nigeria]] |211,400,708 |923,768 |- |[[Republic of the Congo]] |6,142,180 |342,000 |- |[[Rwanda]] |14,569,341 |26,338 |- |[[Senegal]] |16,876,720 |196,722 |- |[[Seychelles]] |119,773 |459 |- |[[Sierra Leone]] |7,557,212 |71,740 |- |[[Somalia]] |11,031,386 |637,657 |- |[[South Africa]] |62,027,503 |1,221,037 |- |[[South Sudan]] |12,575,714 |644,329 |- |[[Sudan]] |40,533,330 |1,886,068 |- |[[São Tomé and Príncipe]] |204,327 |1,001 |- |[[Tanzania]] |57,310,019 |947,303 |- |[[The Gambia]] |2,639,916 |11,300 |- |[[Togo]] |7,797,694 |56,785 |- |[[Tunisia]] |11,565,204 |163,610 |- |[[Uganda]] |47,123,531 |241,038 |- |[[Zambia]] |19,610,769 |752,618 |- |[[Zimbabwe]] |15,178,979 |390,757 |} </div></div> === Top 5 by Population === {| class="wikitable sortable" !Rank !Country !Population |- |1 |[[Nigeria]] |211,400,708 |- |2 |[[Ethiopia]] |128,691,692 |- |3 |[[Egypt]] |114,535,772 |- |4 |[[Democratic Republic of the Congo]] |105,789,731 |- |5 |[[South Africa]] |62,027,503 |} === Top 5 by Area === {| class="wikitable sortable" !Rank !Country !Area (km²) |- |1 |[[Algeria]] |2,381,741 |- |2 |[[Democratic Republic of the Congo]] |2,344,858 |- |3 |[[Sudan]] |1,886,068 |- |4 |[[Libya]] |1,759,541 |- |5 |[[Chad]] |1,284,000 |} === Analysis === The two top-5 lists present distinct narratives regarding regional influence and national importance. Sorting by population highlights Nigeria as the dominant demographic force on the continent, centering the argument about Africa on human capital, labor markets, and urban density. In contrast, sorting by area emphasizes the territorial vastness of Northern and Central African states like Algeria and Chad, framing the continent's importance through land resources and geographical scale. === Data Quality === The raw dataset contains notable anomalies that reflect the complexity of open-source linked data: * '''Duplicate Entries:''' Sudan appeared twice in the raw Wikidata TSV area query with slightly different values (1,886,068 km² and 1,840,687 km²). The higher value was used in the tables above; the duplicate was not carried forward into the published page. * '''Missing Data:''' Not all sovereign states may appear in the combined query if either the population or area property was missing for a specific Wikidata item at the time of the query. * '''Precision Variation:''' Egypt's area is provided with high decimal precision compared to rounded numbers for other states, indicating inconsistent data entry sources. === Reflection === The interactive version gives the reader something the static page cannot: the ability to generate their own argument from the same data. A reader who sorts by area and then by population in succession is performing an act of analysis, not just consumption — they are watching the continent reorganize itself around different claims. The wikilinks transform each country name from a label into a doorway, turning a table about Africa into an entry point into the broader knowledge graph that surrounds it. {{:DesignWriteStudio/SiteElements/Footer}} 5xphogk5rgqr5mezeraj80xmcmydxhv 0 329297 2806698 2026-04-26T19:32:22Z Redwidow448 3044853 Put in my research 2806698 wikitext text/x-wiki Research done by Sarah Jordan as part of the practicum semester (pre-student teaching) at the University of Georgia. The goal of this research was to identify the effects of Problem Based Learning, and see how these effects affected student learning and student discourse. == Introduction: == Throughout my time in the University of Georgia’s mathematics education program, I have noticed a common theme of using strong mathematical tasks to teach. In our first semester, we talked a lot about different types of tasks, the levels of the tasks (whether they were high floor or low ceiling, and types of cognitive demand), and discussed different ideas of what made a task a good task. Then in the second semester, we focused a lot on group work and discussing what makes a task group worthy. Finally, this semester, in addition to trying to put those ideas into motion with the micro-teaching project and in our practicum, I have also been taking a problem solving class that focuses on developing and aiding students in problem solving skills. This is where my interest in problem based learning (PBL) came from, because it felt to me that we had been thinking about PBL since day one of the program. Problem based learning can have varying definitions, but the basic idea behind it is still the same. For the sake of the research, I will use the definition provided in the article ''Teaching Geometry Through Problem Based Learning'' which says the problem based learning is “an instructional approach of curriculum and pedagogy where student learning and content material are constructed (and co-constructed) through the use, facilitation, and experience of contextual problems in a decompartmentalized, threaded topic format in a discussion-based classroom setting where student voice, experience, and prior knowledge are valued” (Schettino, 2012, p. 347). When I saw PBL as an option to research, I knew that I wanted to investigate, even further than we have in our EMAT classes, some of the effects PBL can have on students. More specifically, I wanted to research how problem based learning can benefit student learning and student discourse. From my research, I have found that problem based learning provides students with deeper mathematical conceptual understanding, helps students build autonomy and inner motivation in solving problems, and can have lasting beneficial effects on a student’s mathematical self-concept all of which either increases student learning, student discourse, or both. == Claim #1: Problem based learning increases student learning by providing students with deeper mathematical conceptual understanding. == Though some activities in PBL may have one correct solution and some may have multiple solutions, the act of solving these problems is much more of an open approach than the usual closed approaches that typical math problems may have. It is because of this more open approach in solving the problems that allows students to develop a deep conceptual understanding that provides “them with advantages in a range of assessments and situations” (Boaler, 1998, p. 41). In the article by Jo Boaler (1998) titled ''Open and Closed Mathematics: Student Experiences and Understandings'', Boaler observes two schools in which one has the typical closed textbook teaching approach, and the other has more of an open project-based approach. The school that had an open project approach to learning not only had better test scores than the closed approach but also had a better ability to adapt procedures and ideas in new situations. Thus, these students were able to transfer different ideas to new, unfamiliar problems than the students faced with a more closed teaching approach. This means that the students had a deeper understanding of the mathematical concepts which allowed them to see different mathematical ideas and procedures as tools that can be adapted and used in a variety of ways, rather than only seeing one single use for a procedure (Boaler, 1998). Because PBL also has a more open approach to learning, these conclusions from the article also take place in a PBL approach. Students in an open PBL classroom get the chance to work with and build mathematical concepts themselves, instead of the teacher presenting them the mathematics (Fi & Degner, 2012). Because of this, students can make connections between and understand the ideas behind mathematical concepts. This all means that students are learning mathematical concepts because they are able to use and recall mathematical ideas in new problems and situations, often on their own. In ''Teaching Geometry Through Problem Based Learning'', Schettino (2012) shares another example of students connecting mathematical concepts using their own ideas and preconceptions. The goal of the problem was to introduce and build the student’s understanding of the sine function. The problem asked students to make, with a ruler and protractor, a right triangle that had a hypotenuse of 15 cm and an angle of 27 degrees. Then the students had to see what percentage of the opposite side of the angle was of the hypotenuse. One student had misread the problem and used 10 cm as the hypotenuse measurement and was wondering how she had also gotten the percentage of about 45%. After a bit, another student offered the idea that “she just made a similar triangle like in that problem we did yesterday” (Schettino, 2012, p. 349). This student made a connection between similar triangles, proportionality, and the sine ratio, demonstrating how they had not only learned similar triangles, but had a deep enough understanding of the concept to make a connection to the current mathematical idea. PBL increases a student's learning, as shown in the previous example, by allowing them to make overarching connections between mathematical concepts and develop a better understanding of how mathematics connects with each other, as opposed to the usual closed textbook, unit to unit, classroom. == Claim #2: Problem based learning increases and encourages student learning and discourse by helping students build autonomy in solving problems. == Problem solving skills play a big role in PBL. As mentioned in our previous definition, in PBL students learn and connect mathematical content through problems and tasks, which means that a student’s problem solving skills are constantly used and put to the test. Students must explore the problem, build conjectures, struggle productively, and share their ideas with others to engage in the mathematics and mathematical practices within the problems (Fi & Degner, 2012). When students are exploring the problem and building conjectures, the cognitive load of the problem itself is put on the students, causing the students to have to decide how to go about the problem and what tools or ideas they should use. Students are also having to decide, usually, what ideas may or may not work, and be willing to try a new conjecture when a conjecture or route may not work. This builds the students’ autonomy because they are having to monitor their own thinking and ideas related to the problem and realize when they may need to try a new idea. This autonomy also relates to the idea of productive struggle. With productive struggle, students are building their resilience to struggle while also becoming more aware of when the struggle could be past the productive point. As a student becomes more aware of these points of productive vs unproductive struggle, and builds their resilience, their autonomy and “agency in learning” increases and “independence in problem solving” increases (Schettino, 2012, p. 350). The students know when they may need to take a break or try a new idea without being told to do so the more they engage in problem based learning. This autonomy that the students gain also increases their learning and connecting mathematical ideas. Deciding how to go about a problem, and what ideas could be used demonstrates that a student understands what the problem is asking and is able to pull from previously gained mathematical ideas and make connections about how certain ideas relate to another. For example, in the previous paragraph’s example in the geometry classroom, the student was able to connect similar triangles to the proportionality of sine (Schettino, 2012). This demonstrates that the student connected the proportionality idea of similar triangles to the proportionality of sine, increasing the student’s knowledge of proportionality. The key detail here is that the student had the autonomy to propose the idea after thinking about it for a bit, instead of trying to propose a solution right away, making a connection to an idea that was related, deepening his knowledge of the mathematical concept. Additionally, a student’s autonomy, gained through PBL, increases discourse between students. As students are making conjectures and sharing ideas with other students, in order to solve or begin to solve problems, they are having to actively decide what ideas to share and having to practice their communication skills. Through sharing and conversing with other students to solve a problem, students experience “improved” communication skills and “empowerment in student voice” (Schettino, 2012, p. 350). The students are learning through the problems themselves, meaning that they are also learning from others. The more the students are having and willing to share their ideas, the better the students become at voicing their thoughts and answers. As they always say, practice makes perfect, or in the case of PBL, the more a student shares their ideas, the better they become at communicating said ideas. Furthermore, because students are learning through the problems themselves, with some guidance from the teacher in the discussion and summing up of ideas, student voices become more important to other students because it’s where much of the learning comes from. Since a problem based classroom requires discussion to learn the concepts, and the students influence most of the discussion, a student’s addition to the discussion becomes vital in the learning of other students, thus empowering the students’ voice. == Claim #3: PBL increases and encourages student learning and discourse by building students’ inner motivation in solving problems in math. == In a study discussed in the article ''Assessment of a Problem-Centered Mathematics Program: Third Grade'', it was found that the classes of students that were exposed to problem based learning for one to two years, with two years being the strongest results, the students “believe that it is important to find their own or different ways to solve problems, rather than conform to the method shown by the teacher” and “both project groups are less likely to be motivated to avoid challenging tasks or to best others in order to be successful in mathematics class” (Wood & Sellers, 1996, p. 351). Not only does this demonstrate the previous point of autonomy, because the students wanted to find their own solutions, but this also demonstrates the students’ inner motivation towards math. The students have the motivation to try and solve the problem in a different way, or to at least try to, instead of just copying exactly what the teacher does. They see the importance of observing different solution methods, methods that may include different ways of thinking or maybe even different math concepts, which, in turn, increases their knowledge and learning of the math itself. Moreover, students being motivated to tackle challenging tasks shows an even deeper motivation towards math. In traditional settings, it’s common for students who are faced with a challenging problem to disengage. However, the fact that the students want to actively try challenging themselves with these problems shows how motivated the students are in learning math. While the research from this study focuses mostly on elementary math, if we engage students with mathematics earlier on, problem based learning could have this effect on the inner motivation of secondary students as well because I have seen this in my own practicum experience. During my mentor teacher's quadratic unit, he used a few Desmos lessons and problems to help build student’s understanding behind how and why a quadratic equation is formed and would often hold discussions with the students to clarify and see their ideas. While the students were working on the problems through the many days, I observed multiple groups working together to solve the problem. There were a few occasions where I would see a group of students get an answer right (desmos would tell them if a simple typed in answer was right) to a rather challenging idea, and then immediately wonder why. Because the students, who were eighth graders, would try and figure out why a solution was right, instead of just moving on, demonstrates that the students had this inner motivation to understand the math and think through different solution ideas. This inner motivation gained through PBL increases student learning and discourse because the students are gaining important insights from challenging problems, as well as solving the problems in multiple ways. As mentioned, solving the problem in multiple ways, as well as solving challenging problems, increases a student’s exposure to ways that different ideas or concepts in math can be used to solve problems, building mathematical connections between ideas for students. Likewise, because students are having to share their ideas and the students aren’t inwardly motivated to “best” others to be successful in mathematics, student discourse is not only increased, but also important to the students (Wood & Sellers, 1996, p. 351). The students, with this inner motivation, see the importance of others' contributions and ideas in their learning, and realize that problem based learning is a collaborative effort, meaning that all ideas and contributions are welcome. This increases, and empowers, student discourse because more students are likely to want to share and hear other students' ideas to not only solve the problem, but to learn as well.   == Claim #4: PBL can have lasting beneficial effects on a student’s mathematical self-concept, increasing student learning and student discourse. == A student’s mathematical self-concept is often defined as a student’s belief “of their skills, ability, enjoyment and interest in mathematics” (Erdogan & Sengul, 2014, p. 596). The stronger a student’s mathematical self-concept is, the stronger the student feels about their math skills, ability, and their enjoyment of mathematics. If a student feels confident, and strong, about their problem solving skills, which are heavily used in PBL, the stronger their self-concept is in not only math but in life as well. This is because students feel they can tackle any problem that may arise because they have the skills to solve it without getting too frustrated. In a study focused on assessing how problem based digital learning (PBDL) affects vocational high school students’ mathematical proficiency and creative problem solving skills, the researchers found that PBDL improves not only student’s mathematical proficiency, but also their creative problem solving (Yang et al., 2025). Thus, because creative problem solving allows students to use multiple valid approaches to solve the problems, students are less likely to feel like they failed if their first attempt doesn’t work, which actively builds their mathematical self-concept, and their overall self-concept as well. As previously mentioned, when students have these problem solving skills, the more tools they have to face any math problem given to them, giving them strong beliefs in their math skills and inner motivation to solve the problem. Additionally, since vocational high school students and traditional students “must apply mathematics proficiency and creative problem solving in real and complex industrial contexts,” this means that these problem solving skills developed through PBL are skills that can be attributed to any part of life, and provide the students with a set of important skills to use in their future endeavors (Yang et al., 2025, p. 23792). Hence, PBL can create lasting beneficial effects on a students’ mathematical self-concept, as well as their overall self-concept, since the problem solving skills and the communication skills are tools that students will be able to use as time goes on. This lasting beneficial effect on a student’s self-concept, that PBL has, increases both student learning and discourse. Similar to the reasoning brought up about a student’s inner motivation increasing learning, as a student’s mathematical self-concept increases the more likely the student is to attempt new problems, in ways that both increase their mathematical proficiency and their ability to make connections between ideas. When students are unafraid to try new problems, due to a high mathematical (or general) self-concept, then they are more likely to be exposed to new ideas and math concepts that they can connect to previously known ideas; This not only extends their learning, but also allows students to make connections between concepts, increasing their mathematical proficiency. Moreover, because of the instinctual collaborative nature that both PBDL and PBL hold, students experience increased discourse and collaboration between themselves and other students (Yang et al., 2025). Since PBL and PBDL both rely on class and group discussions to solve the problems, this means that the students, while collaborating, are having to discuss and share their ideas more often than they would if they were just completing worksheets on their own, or in pairs. With a strong self-concept, in these class discussions, students aren’t afraid to not only try different ideas when problem solving, but also to voice their ideas in front of their peers as well. So, if they say an idea to solve a problem to their peers, and it doesn’t end up working or gets debunked rather quickly, the students aren’t stressed because they know their abilities and recognize that it was just an idea. Furthermore, a strong self-concept also allows students to not be afraid to make conjectures that may not be completely thought through, encouraging further collaboration and discussions between students because they will spend time thinking through the conjecture. Hence, a student’s mathematical self-concept, made stronger through PBL, increases student learning and student discourse. == Conclusion: == Considering the fact that PBL increases student learning and student discourse in multiple ways—such as providing students with deeper mathematical conceptual understanding, helping students build autonomy and inner motivation in solving problems, and having lasting beneficial effects on a student’s mathematical self-concept—I am more convinced that I should incorporate problem based learning in my future classroom. As mentioned in the introduction, I already had an impression of how important a mathematical task can be for student learning. However, my research has indicated just how important learning through tasks and problems can be for students. I want my students to not only learn math, but to be able to make important mathematical connections, while also building an appreciation for math itself. It’s evident that PBL is a teaching method that allows students to make these overarching mathematical connections, either themselves or through collaborative discussion, deepening the student’s understanding in mathematics, and the class’ understanding for mathematics as well. I want to show my students that it’s okay to make mistakes and it’s important to be willing to take risks by sharing your ideas and collaborating with others. It’s through these discussions that, as demonstrated in my research, students are able to learn from one another and make these important mathematical connections. That being said, it may be hard to instill these ideas of collaboration at first, but as I intend to use frequent lessons based in problem based learning, these skills will build for the students over time and, evidently, can have a major impact on my students beliefs in their mathematical ability, and can foster in-depth learning, understanding, and appreciation for mathematics. == Citations/Bibliography: == Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. ''Journal for Research in Mathematics Education'', ''29''(1), 41–62. <nowiki>https://doi.org/10.2307/749717</nowiki> Erdogan, F., & Sengul, S. (2014). A study on the elementary school students' mathematics self-concept. ''Procedia - Social and Behavioral Sciences'', ''152'', 596-601. <nowiki>https://doi.org/10.1016/j.sbspro.2014.09.249</nowiki> Fi, C. D., & Degner, K. M. (2012). Teaching through problem solving. ''The Mathematics Teacher'', ''105''(6), 455-459. <nowiki>https://doi.org/10.5951/mathteacher.105.6.0455</nowiki> Schettino, C. (2012). Teaching geometry through problem-based learning. ''The Mathematics Teacher'', ''105''(5), 346–351. <nowiki>https://doi.org/10.5951/mathteacher.105.5.0346</nowiki> Wood, T., & Sellers, P. (1996). Assessment of a problem-centered mathematics program:Third grade. ''Journal for Research in Mathematics Education'', ''27''(3), 337–353. <nowiki>https://doi.org/10.2307/749368</nowiki> Yang, S., Zhu, S., Qin, W., Mai, Y., Guo, Q., Li, H. (2025) Assessing the impact of problem-based digital learning on mathematics proficiency and creative problem solving for vocational high students. ''Education and Information Technol''ogies 30, 23791–23816. <nowiki>https://doi.org/10.1007/s10639-025-13710-6</nowiki> 80k7g4qrbokfodl4horo4qc2i73tizy 2806699 2806698 2026-04-26T19:39:13Z Redwidow448 3044853 added poster 2806699 wikitext text/x-wiki [[File:Poster For Research.png|center|thumb|620x620px|This was the poster created to present the following research.]] Research done by Sarah Jordan as part of the practicum semester (pre-student teaching) at the University of Georgia. The goal of this research was to identify the effects of Problem Based Learning, and see how these effects affected student learning and student discourse. == Introduction: == Throughout my time in the University of Georgia’s mathematics education program, I have noticed a common theme of using strong mathematical tasks to teach. In our first semester, we talked a lot about different types of tasks, the levels of the tasks (whether they were high floor or low ceiling, and types of cognitive demand), and discussed different ideas of what made a task a good task. Then in the second semester, we focused a lot on group work and discussing what makes a task group worthy. Finally, this semester, in addition to trying to put those ideas into motion with the micro-teaching project and in our practicum, I have also been taking a problem solving class that focuses on developing and aiding students in problem solving skills. This is where my interest in problem based learning (PBL) came from, because it felt to me that we had been thinking about PBL since day one of the program. Problem based learning can have varying definitions, but the basic idea behind it is still the same. For the sake of the research, I will use the definition provided in the article ''Teaching Geometry Through Problem Based Learning'' which says the problem based learning is “an instructional approach of curriculum and pedagogy where student learning and content material are constructed (and co-constructed) through the use, facilitation, and experience of contextual problems in a decompartmentalized, threaded topic format in a discussion-based classroom setting where student voice, experience, and prior knowledge are valued” (Schettino, 2012, p. 347). When I saw PBL as an option to research, I knew that I wanted to investigate, even further than we have in our EMAT classes, some of the effects PBL can have on students. More specifically, I wanted to research how problem based learning can benefit student learning and student discourse. From my research, I have found that problem based learning provides students with deeper mathematical conceptual understanding, helps students build autonomy and inner motivation in solving problems, and can have lasting beneficial effects on a student’s mathematical self-concept all of which either increases student learning, student discourse, or both. == Claim #1: Problem based learning increases student learning by providing students with deeper mathematical conceptual understanding. == Though some activities in PBL may have one correct solution and some may have multiple solutions, the act of solving these problems is much more of an open approach than the usual closed approaches that typical math problems may have. It is because of this more open approach in solving the problems that allows students to develop a deep conceptual understanding that provides “them with advantages in a range of assessments and situations” (Boaler, 1998, p. 41). In the article by Jo Boaler (1998) titled ''Open and Closed Mathematics: Student Experiences and Understandings'', Boaler observes two schools in which one has the typical closed textbook teaching approach, and the other has more of an open project-based approach. The school that had an open project approach to learning not only had better test scores than the closed approach but also had a better ability to adapt procedures and ideas in new situations. Thus, these students were able to transfer different ideas to new, unfamiliar problems than the students faced with a more closed teaching approach. This means that the students had a deeper understanding of the mathematical concepts which allowed them to see different mathematical ideas and procedures as tools that can be adapted and used in a variety of ways, rather than only seeing one single use for a procedure (Boaler, 1998). Because PBL also has a more open approach to learning, these conclusions from the article also take place in a PBL approach. Students in an open PBL classroom get the chance to work with and build mathematical concepts themselves, instead of the teacher presenting them the mathematics (Fi & Degner, 2012). Because of this, students can make connections between and understand the ideas behind mathematical concepts. This all means that students are learning mathematical concepts because they are able to use and recall mathematical ideas in new problems and situations, often on their own. In ''Teaching Geometry Through Problem Based Learning'', Schettino (2012) shares another example of students connecting mathematical concepts using their own ideas and preconceptions. The goal of the problem was to introduce and build the student’s understanding of the sine function. The problem asked students to make, with a ruler and protractor, a right triangle that had a hypotenuse of 15 cm and an angle of 27 degrees. Then the students had to see what percentage of the opposite side of the angle was of the hypotenuse. One student had misread the problem and used 10 cm as the hypotenuse measurement and was wondering how she had also gotten the percentage of about 45%. After a bit, another student offered the idea that “she just made a similar triangle like in that problem we did yesterday” (Schettino, 2012, p. 349). This student made a connection between similar triangles, proportionality, and the sine ratio, demonstrating how they had not only learned similar triangles, but had a deep enough understanding of the concept to make a connection to the current mathematical idea. PBL increases a student's learning, as shown in the previous example, by allowing them to make overarching connections between mathematical concepts and develop a better understanding of how mathematics connects with each other, as opposed to the usual closed textbook, unit to unit, classroom. == Claim #2: Problem based learning increases and encourages student learning and discourse by helping students build autonomy in solving problems. == Problem solving skills play a big role in PBL. As mentioned in our previous definition, in PBL students learn and connect mathematical content through problems and tasks, which means that a student’s problem solving skills are constantly used and put to the test. Students must explore the problem, build conjectures, struggle productively, and share their ideas with others to engage in the mathematics and mathematical practices within the problems (Fi & Degner, 2012). When students are exploring the problem and building conjectures, the cognitive load of the problem itself is put on the students, causing the students to have to decide how to go about the problem and what tools or ideas they should use. Students are also having to decide, usually, what ideas may or may not work, and be willing to try a new conjecture when a conjecture or route may not work. This builds the students’ autonomy because they are having to monitor their own thinking and ideas related to the problem and realize when they may need to try a new idea. This autonomy also relates to the idea of productive struggle. With productive struggle, students are building their resilience to struggle while also becoming more aware of when the struggle could be past the productive point. As a student becomes more aware of these points of productive vs unproductive struggle, and builds their resilience, their autonomy and “agency in learning” increases and “independence in problem solving” increases (Schettino, 2012, p. 350). The students know when they may need to take a break or try a new idea without being told to do so the more they engage in problem based learning. This autonomy that the students gain also increases their learning and connecting mathematical ideas. Deciding how to go about a problem, and what ideas could be used demonstrates that a student understands what the problem is asking and is able to pull from previously gained mathematical ideas and make connections about how certain ideas relate to another. For example, in the previous paragraph’s example in the geometry classroom, the student was able to connect similar triangles to the proportionality of sine (Schettino, 2012). This demonstrates that the student connected the proportionality idea of similar triangles to the proportionality of sine, increasing the student’s knowledge of proportionality. The key detail here is that the student had the autonomy to propose the idea after thinking about it for a bit, instead of trying to propose a solution right away, making a connection to an idea that was related, deepening his knowledge of the mathematical concept. Additionally, a student’s autonomy, gained through PBL, increases discourse between students. As students are making conjectures and sharing ideas with other students, in order to solve or begin to solve problems, they are having to actively decide what ideas to share and having to practice their communication skills. Through sharing and conversing with other students to solve a problem, students experience “improved” communication skills and “empowerment in student voice” (Schettino, 2012, p. 350). The students are learning through the problems themselves, meaning that they are also learning from others. The more the students are having and willing to share their ideas, the better the students become at voicing their thoughts and answers. As they always say, practice makes perfect, or in the case of PBL, the more a student shares their ideas, the better they become at communicating said ideas. Furthermore, because students are learning through the problems themselves, with some guidance from the teacher in the discussion and summing up of ideas, student voices become more important to other students because it’s where much of the learning comes from. Since a problem based classroom requires discussion to learn the concepts, and the students influence most of the discussion, a student’s addition to the discussion becomes vital in the learning of other students, thus empowering the students’ voice. == Claim #3: PBL increases and encourages student learning and discourse by building students’ inner motivation in solving problems in math. == In a study discussed in the article ''Assessment of a Problem-Centered Mathematics Program: Third Grade'', it was found that the classes of students that were exposed to problem based learning for one to two years, with two years being the strongest results, the students “believe that it is important to find their own or different ways to solve problems, rather than conform to the method shown by the teacher” and “both project groups are less likely to be motivated to avoid challenging tasks or to best others in order to be successful in mathematics class” (Wood & Sellers, 1996, p. 351). Not only does this demonstrate the previous point of autonomy, because the students wanted to find their own solutions, but this also demonstrates the students’ inner motivation towards math. The students have the motivation to try and solve the problem in a different way, or to at least try to, instead of just copying exactly what the teacher does. They see the importance of observing different solution methods, methods that may include different ways of thinking or maybe even different math concepts, which, in turn, increases their knowledge and learning of the math itself. Moreover, students being motivated to tackle challenging tasks shows an even deeper motivation towards math. In traditional settings, it’s common for students who are faced with a challenging problem to disengage. However, the fact that the students want to actively try challenging themselves with these problems shows how motivated the students are in learning math. While the research from this study focuses mostly on elementary math, if we engage students with mathematics earlier on, problem based learning could have this effect on the inner motivation of secondary students as well because I have seen this in my own practicum experience. During my mentor teacher's quadratic unit, he used a few Desmos lessons and problems to help build student’s understanding behind how and why a quadratic equation is formed and would often hold discussions with the students to clarify and see their ideas. While the students were working on the problems through the many days, I observed multiple groups working together to solve the problem. There were a few occasions where I would see a group of students get an answer right (desmos would tell them if a simple typed in answer was right) to a rather challenging idea, and then immediately wonder why. Because the students, who were eighth graders, would try and figure out why a solution was right, instead of just moving on, demonstrates that the students had this inner motivation to understand the math and think through different solution ideas. This inner motivation gained through PBL increases student learning and discourse because the students are gaining important insights from challenging problems, as well as solving the problems in multiple ways. As mentioned, solving the problem in multiple ways, as well as solving challenging problems, increases a student’s exposure to ways that different ideas or concepts in math can be used to solve problems, building mathematical connections between ideas for students. Likewise, because students are having to share their ideas and the students aren’t inwardly motivated to “best” others to be successful in mathematics, student discourse is not only increased, but also important to the students (Wood & Sellers, 1996, p. 351). The students, with this inner motivation, see the importance of others' contributions and ideas in their learning, and realize that problem based learning is a collaborative effort, meaning that all ideas and contributions are welcome. This increases, and empowers, student discourse because more students are likely to want to share and hear other students' ideas to not only solve the problem, but to learn as well.   == Claim #4: PBL can have lasting beneficial effects on a student’s mathematical self-concept, increasing student learning and student discourse. == A student’s mathematical self-concept is often defined as a student’s belief “of their skills, ability, enjoyment and interest in mathematics” (Erdogan & Sengul, 2014, p. 596). The stronger a student’s mathematical self-concept is, the stronger the student feels about their math skills, ability, and their enjoyment of mathematics. If a student feels confident, and strong, about their problem solving skills, which are heavily used in PBL, the stronger their self-concept is in not only math but in life as well. This is because students feel they can tackle any problem that may arise because they have the skills to solve it without getting too frustrated. In a study focused on assessing how problem based digital learning (PBDL) affects vocational high school students’ mathematical proficiency and creative problem solving skills, the researchers found that PBDL improves not only student’s mathematical proficiency, but also their creative problem solving (Yang et al., 2025). Thus, because creative problem solving allows students to use multiple valid approaches to solve the problems, students are less likely to feel like they failed if their first attempt doesn’t work, which actively builds their mathematical self-concept, and their overall self-concept as well. As previously mentioned, when students have these problem solving skills, the more tools they have to face any math problem given to them, giving them strong beliefs in their math skills and inner motivation to solve the problem. Additionally, since vocational high school students and traditional students “must apply mathematics proficiency and creative problem solving in real and complex industrial contexts,” this means that these problem solving skills developed through PBL are skills that can be attributed to any part of life, and provide the students with a set of important skills to use in their future endeavors (Yang et al., 2025, p. 23792). Hence, PBL can create lasting beneficial effects on a students’ mathematical self-concept, as well as their overall self-concept, since the problem solving skills and the communication skills are tools that students will be able to use as time goes on. This lasting beneficial effect on a student’s self-concept, that PBL has, increases both student learning and discourse. Similar to the reasoning brought up about a student’s inner motivation increasing learning, as a student’s mathematical self-concept increases the more likely the student is to attempt new problems, in ways that both increase their mathematical proficiency and their ability to make connections between ideas. When students are unafraid to try new problems, due to a high mathematical (or general) self-concept, then they are more likely to be exposed to new ideas and math concepts that they can connect to previously known ideas; This not only extends their learning, but also allows students to make connections between concepts, increasing their mathematical proficiency. Moreover, because of the instinctual collaborative nature that both PBDL and PBL hold, students experience increased discourse and collaboration between themselves and other students (Yang et al., 2025). Since PBL and PBDL both rely on class and group discussions to solve the problems, this means that the students, while collaborating, are having to discuss and share their ideas more often than they would if they were just completing worksheets on their own, or in pairs. With a strong self-concept, in these class discussions, students aren’t afraid to not only try different ideas when problem solving, but also to voice their ideas in front of their peers as well. So, if they say an idea to solve a problem to their peers, and it doesn’t end up working or gets debunked rather quickly, the students aren’t stressed because they know their abilities and recognize that it was just an idea. Furthermore, a strong self-concept also allows students to not be afraid to make conjectures that may not be completely thought through, encouraging further collaboration and discussions between students because they will spend time thinking through the conjecture. Hence, a student’s mathematical self-concept, made stronger through PBL, increases student learning and student discourse. == Conclusion: == Considering the fact that PBL increases student learning and student discourse in multiple ways—such as providing students with deeper mathematical conceptual understanding, helping students build autonomy and inner motivation in solving problems, and having lasting beneficial effects on a student’s mathematical self-concept—I am more convinced that I should incorporate problem based learning in my future classroom. As mentioned in the introduction, I already had an impression of how important a mathematical task can be for student learning. However, my research has indicated just how important learning through tasks and problems can be for students. I want my students to not only learn math, but to be able to make important mathematical connections, while also building an appreciation for math itself. It’s evident that PBL is a teaching method that allows students to make these overarching mathematical connections, either themselves or through collaborative discussion, deepening the student’s understanding in mathematics, and the class’ understanding for mathematics as well. I want to show my students that it’s okay to make mistakes and it’s important to be willing to take risks by sharing your ideas and collaborating with others. It’s through these discussions that, as demonstrated in my research, students are able to learn from one another and make these important mathematical connections. That being said, it may be hard to instill these ideas of collaboration at first, but as I intend to use frequent lessons based in problem based learning, these skills will build for the students over time and, evidently, can have a major impact on my students beliefs in their mathematical ability, and can foster in-depth learning, understanding, and appreciation for mathematics. == Citations/Bibliography: == Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. ''Journal for Research in Mathematics Education'', ''29''(1), 41–62. <nowiki>https://doi.org/10.2307/749717</nowiki> Erdogan, F., & Sengul, S. (2014). A study on the elementary school students' mathematics self-concept. ''Procedia - Social and Behavioral Sciences'', ''152'', 596-601. <nowiki>https://doi.org/10.1016/j.sbspro.2014.09.249</nowiki> Fi, C. D., & Degner, K. M. (2012). Teaching through problem solving. ''The Mathematics Teacher'', ''105''(6), 455-459. <nowiki>https://doi.org/10.5951/mathteacher.105.6.0455</nowiki> Schettino, C. (2012). Teaching geometry through problem-based learning. ''The Mathematics Teacher'', ''105''(5), 346–351. <nowiki>https://doi.org/10.5951/mathteacher.105.5.0346</nowiki> Wood, T., & Sellers, P. (1996). Assessment of a problem-centered mathematics program:Third grade. ''Journal for Research in Mathematics Education'', ''27''(3), 337–353. <nowiki>https://doi.org/10.2307/749368</nowiki> Yang, S., Zhu, S., Qin, W., Mai, Y., Guo, Q., Li, H. (2025) Assessing the impact of problem-based digital learning on mathematics proficiency and creative problem solving for vocational high students. ''Education and Information Technol''ogies 30, 23791–23816. <nowiki>https://doi.org/10.1007/s10639-025-13710-6</nowiki> khqen7ceo7qbdtekstwr55wgqh7wuc7 0 329298 2806741 2026-04-27T03:54:40Z OhanaUnited 18921 create 2806741 wikitext text/x-wiki <noinclude>{{WikiJHum top menu}} {{col-begin}} {{col-break}} </noinclude> {{Issue header|year = 2026 |volume = 7 |issue = 1 |current=true}} {{Article info |Q = Q139393332 |image=Wikipedia and WikiProjects in the Focus of Research (horizontal).png}} <noinclude>{{col-break}} {{WikiJHum right menu}} {{col-end}} [[category:{{ROOTPAGENAME}} issues]] </noinclude> rb925w57u2ukui4sv47nseej83c2435 0 329299 2806750 2026-04-27T04:15:15Z OhanaUnited 18921 XML created with data from [[template:article_info]] 2806750 wikitext text/x-wiki {{Autogenerate_xml}} 9ydjcj3h2hxjmnhnepnvvslgiatxxrk 0 329300 2806751 2026-04-27T04:18:10Z OhanaUnited 18921 XML created with data from [[template:article_info]] 2806751 wikitext text/x-wiki {{Autogenerate_xml}} 9ydjcj3h2hxjmnhnepnvvslgiatxxrk 0 329301 2806756 2026-04-27T04:23:35Z OhanaUnited 18921 XML created with data from [[template:article_info]] 2806756 wikitext text/x-wiki {{Autogenerate_xml}} 9ydjcj3h2hxjmnhnepnvvslgiatxxrk 0 329302 2806764 2026-04-27T04:38:27Z OhanaUnited 18921 XML created with data from [[template:article_info]] 2806764 wikitext text/x-wiki {{Autogenerate_xml}} 9ydjcj3h2hxjmnhnepnvvslgiatxxrk 6 329304 2806774 2026-04-27T10:03:08Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260427 - 20260425) |Source={{own|Young1lim}} |Date=2026-04-27 |Author=Young W. 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Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} e1z86oevtyjy6muzl9t0ti3ozq6xmeo 6 329306 2806779 2026-04-27T10:22:46Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260427 - 20260425) |Source={{own|Young1lim}} |Date=2026-04-27 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2806779 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260427 - 20260425) |Source={{own|Young1lim}} |Date=2026-04-27 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 9z71mat5s7xpzv65fhoorayoi0pxi8n 0 329307 2806781 2026-04-27T11:22:55Z Projet PEP 3002502 Created page with "== Starting activity == '''Activity A''' Visualise your own linguistic constellations by following the steps below: [[File:Dominant language constellation - Activity 1.png|alt=Dominant language constellation - Activity 1|thumb|Figure 1 - Dominant language constellation - Activity 1]] 1. Copy the exercise onto a worksheet (cf. Figure 1). 2. Name each language, dialect or variety that you use in your daily life (write them in the coloured bubbles). 3. Complete the sen..." 2806781 wikitext text/x-wiki == Starting activity == '''Activity A''' Visualise your own linguistic constellations by following the steps below: [[File:Dominant language constellation - Activity 1.png|alt=Dominant language constellation - Activity 1|thumb|Figure 1 - Dominant language constellation - Activity 1]] 1. Copy the exercise onto a worksheet (cf. Figure 1). 2. Name each language, dialect or variety that you use in your daily life (write them in the coloured bubbles). 3. Complete the sentences in the boxes. For example: * I use it at home with my grandmother. * For example, I say: “Wie geht es dir?”  (= How are you?) 4. List your languages, dialects and varieties and write the number in the central square '''Activity B''' [[File:Dominant language constellation - Activity 2.png|alt=Dominant language constellation - Activity 2|thumb|Figure 2 - Dominant language constellation - Activity 2]] 1. Copy the exercise onto a sheet of paper (cf. Figure 2). 2.&nbsp;Place the languages / dialects / varieties you listed earlier into the three circles below * Circle A: Languages, dialects and varieties that you use frequently. * Circle B: Languages, dialects and varieties that you use occasionally. * Circle C: Languages, dialects and varieties that you rarely use, but which you sometimes hear or know people who speak them. 3.&nbsp;Write down why you have placed them in each circle. '''Activity C''' Think about the results: * How proficient are you in the languages, dialects and varieties of the circles A, B and C? To what extent can you understand them (in writing and orally), write them and speak them?  * Does anything surprise you? What might that mean? (Exercise adapted from Xu & Krulatz, 2024, p. 283) == Objectives == By the end of this section, you should be able to… * distinguish the dominant linguistic constellation from other concepts related to plurilingualism/multilingualism (e.g. the linguistic repertoire) * describe and analyse the DLC (and its development) * apply the DLC theory in an educational context == Keywords == plurilinguism, multilinguism, linguistic repertoire, communicative practices, institutional linguistic policy == Pre-requisites == There are no prerequisites, but an understanding of concepts such as linguistic repertoire or ''translanguaging'' will help you gain a deeper understanding of the content. == Introduction == The Dominant Language Constellation (DLC) was proposed by Larissa Aronin and her colleagues (2006, 2016, 2019; Aronin & Singleton 2012) as a way of addressing the dynamic diversity of globalised language practices.  A DLC refers to the group of languages, dialects and/or varieties most relevant to an individual or a group for meeting all their linguistic needs in a multilingual environment. The components of a DLC function as a unit. In other words, the DLC describes the inner circle of personal languages that ‘fulfil the most vital functions of language” (Aronin, 2016, p.147). == History of the concept == The DLC is one of many critical responses to the challenge  to the historically and culturally entrenched notion that monolingualism is ‘natural’ or preferable, whether at the level of individuals, groups or nations. This kind of monolingual ideology has been widely propagated by nationalism, Romanticism and state education systems.<blockquote>''A short historical example:'' ''A key example of this link between language policy and nation-building is Talleyrand’s speech in 1791, during the French Revolution, in which he argued for the exclusive use of the national language in schools. Dialects and regional languages were described in it as “corrupted languages”, associated with a “feudal order that must be abolished” (Talleyrand-Périgord, 1791, p. 472, our translation).''</blockquote>Even today, monolingualism remains a deeply entrenched social and political system, underpinned by nationalist ideologies, notions of modernised rationality and efficiency-driven approaches (one might cite, for example, certain language policies in the United States under the Trump administration, such as the designation of English as the official language of the United States (White House online, March 2025). Linguistic practices, however, are diverse and multilingual, and have been so long before globalisation. DLC, therefore, reflects the plurilingual realities experienced in everyday life. By identifying the languages actually used in the various contexts of an individual’s life, DLC proposes a model of plurilingual identity, distributed across different domains of use and rooted in concrete communication practices. This is a relatively recent concept, which has become particularly prominent in the field of multilingualism studies since the 21st century. Examples of edited volumes devoted to DLCs include the 2020 collection ''Dominant Language Constellations: A New Perspective on Multilingualism'', edited by Joseph Lo Bianco and Larissa Aronin. Four further volumes have since followed (Aronin & Vetter, 2021; Aronin & Melo-Pfeifer, 2023; Aronin & Vetter, 2025), all available on the website dedicated to the CLD approach (Dominant Language Constellations (DLC)). == Definition == Given that DLC focuses on language practices, it is an inherently pluralistic and dynamic concept, based on the idea that contemporary social practices draw upon multiple modes of communication, different forms of literacy and a variety of languages (Aronin & Vetter, 2021, p. 7).  To put it simply, the following brief definition could be proposed: CLD refers to the set of languages—often three—that play a particularly important role for an individual or a group in order to meet their linguistic needs in a multilingual environment (Lo Bianco & Aronin, 2020, pp. 15-18). At this point, it is necessary to distinguish DLC from the concept of the linguistic repertoire, which has been used to refer to the totality of languages, dialects, styles, registers, codes and routines that characterise interaction in everyday life (Gumperz, 1968; Busch, 2017). Whilst the linguistic repertoire aims to account for the entirety of an individual’s linguistic experiences, the DLC refers exclusively to the components considered most essential (Lo Bianco & Aronin, 2020, pp. 15–18). The two concepts can thus be regarded as complementary. Furthermore, a constellation such as that proposed by DLC always forms within a specific multilingual context, and as this context may change throughout life, the DLC may also change. It can evolve according to geographical, social or cultural frameworks – a diversity clearly highlighted by international studies on local language practices (Vetter, 2024, pp. 229–231). The characteristics of a DLC are not, however, limited to the sum of the languages, dialects and varieties that comprise it.  It must also be regarded as a unit that extends beyond the sum of its parts (which, in any case, cannot be viewed as strictly separate from one another). == Conceptions == Currently, DLC is emerging as a new perspective on multilingualism (Lo Bianco & Aronin 2020) and research method to explore it (Aronin 2019), reflecting the many developments in the field. To clarify these two perspectives, let’s look at them in more detail: * As an innovative approach to multilingualism, it is used to study complex linguistic situations. One example is Karpava’s (2020) study of the Russian community in Cyprus, which highlights the individual and social factors that influence the dynamics of DLCs (composed of the same languages). * As a research method, DLC opens the door to new research questions, as demonstrated by Aronin (2019, 19f). To give an example, DLC can serve as a starting point for describing the linguistic composition of a country, an organisation or the world. Questions arise regarding languages leaving the DLC or returning to it. Just as with conceptual perspectives, the themes addressed by research on DLC have diversified over time and cover a wide range of issues, from multilingual syntactic development (Fernández-Berkes & Flynn, 2019) to educational contexts (Björklund et al., 2019). Others focus on the impact of identity, emotions and attitudes in different minority and majority contexts (Nightingale, 2020; Vetter, 2024). Language policy in the broadest sense is one of the areas in which DLC research yields particularly promising results: a DLC perspective opens up interesting avenues for understanding institutional language policy, particularly in the field of education, where it proves to be a powerful tool for raising awareness and fostering critical reflection (Vetter, 2021). On a more general political level, DLC encourages the involvement of language activists, policy-makers and researchers in a dialogue on appropriate language policy. It can thus serve as a framework for language policy in multilingual communities facing controversial political directions. == Take home messages == The concept of DLC encompasses only the most significant languages, dialects and varieties through which an individual or group can meet their communicative needs. It is, therefore, a concept rooted in practical communication and real needs. The DLC represents a critical response to the notion of ‘natural’ or preferred monolingualism and forms part of modern theories on multilingualism and calls for multicultural practices. It can be interpreted as an innovative perspective to understand multilingualism and a research method, showing particular promise for institutional language policy and the field of education. == Self-assessment == * How can the concept of the Dominant Linguistic Constellation (DLC) be defined? * What is the difference between the concept of a repertoire and the DLC? * What topics can be addressed by research into DLCs? == Further readings == * Official website of the DLC-approach: <nowiki>https://www.dominant-language-constellations.com/</nowiki> * Aronin, L. (2019). Challenges of multilingual education : streamlining affordances through Dominant Language Constellations. ''Stellenbosch Papers in Linguistics Plus'', ''2019''(58), 235–256. https://doi.org/10.5842/58-0-845 * Aronin, L. & Melo-Pfeifer, S. (2023). ''Language Awareness and Identity. Insights via Dominant Language Constellation Approach''. https://doi.org/10.1007/978-3-031-37027-4. * Aronin, L. & Vetter, E. (Eds.) (2025 in print). ''Dominant Language Constellations for Teachers: A practical dimension''.  Springer. * Vetter, E., & Jessner, U. (2019). ''International research on multilingualism: breaking with the monolingual perspective''. Springer. == References == Aronin, L. (2006). Dominant language constellations: An approach to multilingualism studies. In M. Ó Laoire (Ed.), ''Multilingualism in educational settings'' (pp. 140–159). Schneider Publications. Aronin, L. & Singleton, D. (2012). ''Multilingualism''. John Benjamins. Aronin, L. (2016). Multicompetence and dominant language constellation. In V. Cook &  Li Wei (Eds.), ''The Cambridge Handbook of Linguistic Multicompetence'' (pp.142-163). Cambridge University Press. Aronin, L. (2019). Dominant language constellation as a method of research. In E. Vetter & U. Jessner (Eds.), ''International research on multilingualism. Breaking with the monolingual perspective'' (pp. 13–26). Springer. Aronin, L., & Vetter, E. (2021). ''Dominant Language Constellations Approach in Education and Language Acquisition'' (1st Edition 2021, Vol. 51). Springer. https://doi.org/10.1007/978-3-030-70769-9 Björklund, S., Björklund, M. & Sjöholm, K. (2020). Societal Versus Individual Patterns of DLCs in a Finnish Educational Context – Present State and Challenges for the Future. In J. Lo Bianco, L. Aronin (Eds.), ''Dominant Language Constellations: A New Perspective on Multilingualism'' (pp. 97–115). Springer. Busch, B. (2017). Expanding the Notion of the Linguistic Repertoire: On the Concept of Spracherleben —The Lived Experience of Language. ''Applied Linguistics'', ''38''(3),  340-358. https://doi.org/10.1093/applin/amv030 Fernández-Berkes, É. & Flynn, S. (2020). Where DLC Meets Multilingual Syntactic Development. In J. Lo Bianco, L. Aronin (Eds.), ''Dominant Language Constellations: A New Perspective on'' ''Multilingualism'' (pp. 57-74). Springer. Gumperz, J. J. (1968). The speech community. In Sills, D. L. & Merton, R. K. (Eds.), ''International encyclopedia of the social sciences'' (pp. 381-386). Macmillan Company & the Free Press. Karpava, S. (2020). Dominant Language Constellations of Russian Speakers in Cyprus. ​​In J. Lo Bianco & L. Aronin (Eds.), ''Dominant language constellations. A new perspective on multilingualism'' (pp. 228-257). Springer.   Lo Bianco, J., & Aronin, L. (Eds.) (2020). ''Dominant language constellations. A new perspective on multilingualism''. Springer. Nightingale, R. (2020). A Dominant Language Constellations Case Study on Language Use and the Affective Domain. In: ''Dominant language constellations. A new perspective'' ''on multilingualism'' (pp. 231-259). Springer. OJ  2018/C 189/01, Recommandation du Conseil du 22 mai 2018 relative aux compétences clés pour l’éducation et la formation tout au long de la vie. https://eur-lex.europa.eu/legal-content/FR/TXT/PDF/?uri=CELEX:32018H0604(01) Talleyrand-Périgord, C. M. de. (1791). Rapport par M. Talleyrand-Périgord, ancien évêque d’Autun, sur l’instruction publique, en annexe de la séance du 10 septembre 1791. In ''Archives Parlementaires de 1787 à 1860.'' Tome XXX (pp. 447-480). https://www.persee.fr/doc/arcpa_0000-0000_1888_num_30_1_12472_t1_0447_0000_8 The White House. (March 2025). Designating English as the Official Language of The United States. https://www.whitehouse.gov/presidential-actions/2025/03/designating-english-as-the-official-language-of-the-united-states/ UNESCO.  (2025).  ''Les langues comptent : orientations mondiales pour l’éducation multilingue''. https://doi.org/10.54675/UTXF6991 Vetter, E. (2021). Language Education Policy Through a DLC Lens: The Case of Urban Multilingualism. In Aronin, L., & Vetter, E. (Eds.) ''Dominant Language Constellations Approach in Education and Language Acquisition.'' Springer. Vetter, E. (2024). Dominant Instead of Hidden? A Critical Discussion on a European DLC Including Endangered Languages. In ''Modern Approaches to Researching Multilingualism'' (pp. 227–247). Springer. https://doi.org/10.1007/978-3-031-52371-7_14 Xu, Y., Krulatz, A., Gabryś-Barker, D., & Vetter, E. (2024). Employing Dominant Language Constellation in Teacher Professional Development: The Impact on EAL Teachers’ Beliefs, Practices, and Multilingual Identity. In ''Modern Approaches to Researching Multilingualism'' (pp. 271–293). Springer. https://doi.org/10.1007/978-3-031-52371-7_16 g6nt4lvhxchgoupt4kzpyc0zsiy786s