Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.12 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 2819691 2819639 2026-07-28T16:22:38Z Koavf 147 Reverted edits by [[Special:Contributions/~2026-41713-51|~2026-41713-51]] ([[User_talk:~2026-41713-51|talk]]) to last version by [[User:Pine|Pine]] using [[Wikiversity:Rollback|rollback]] 2819376 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) : Thanks for creating this draft. I've made some changes (including moving back to a one month response timeframe) and moved it from draft to proposed policy. : Based on the discussion on the talk page, I think it is close to ready (or ready) to be formally proposed as a policy (by adding it to the sitenotice) to allow wider discussion and hopefully adoption. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:45, 21 July 2026 (UTC) == Proposed user group and/or possible policy changes == {{tracked|T430416|fixed}} I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) :::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:14, 18 June 2026 (UTC) ::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC) == Main page titles == Currently, the title says "Wikiversity:Main Page", but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one): # Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.) # Option 2: Modify [[MediaWiki:Mainpage-title]] to <code>Welcome to Wikiversity</code> (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to <code><nowiki>Welcome to Wikiversity, $1!</nowiki></code>; the latter would display to me as <code>Welcome to Wikiversity, Codename Noreste!</code> Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC) : Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC) :I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:10, 24 June 2026 (UTC) : I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:04, 25 June 2026 (UTC) : {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC) : Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:29, 2 July 2026 (UTC) == Wiki x AI preconference day @ Wikimania == There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. * If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. * If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? Cheers, <span style="padding:0 2px 0 2px;background-color:white;color:#bbb;">&ndash;[[User:Sj|SJ]][[User Talk:Sj|<span style="color:#ff9900;">+</span>]]</span> 23:12, 20 June 2026 (UTC) and Alaexis<br>{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&diff=2816357&oldid=2807052}} == RFC about AI-generated content in Wikimedia Commons == You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC) <!-- 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=30513860 --> == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == <section begin="Message"/> '''Legal & Safety Contacts''' Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]]. <section end="Message"/> -- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC) <!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&oldid=30731267 --> == Preparing manuscript for submission to the ''WikiJournal of Humanities'' == I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''. I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals. The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them. Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC) == I could probably save many kilobytes by compressing my LLM chat history == I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new "method" looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become "secondary citizens". [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC) :{{replyto|ThinkingScience}} I do not see why not. People should be able to look in your page's history for LLM chats before your compression. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 02:34, 14 July 2026 (UTC) ::That's good. Then an admin if they have been instructed to compress...then they can delete the edit history perhaps and save space, then it's up to the admin and I don't need to worry about taking up too many resources. That's good! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 07:50, 14 July 2026 (UTC) == July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual. #'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 03:51, 13 July 2026 (UTC) :@[[User:Pine|Pine]], Some wikimedians who may contribute less popular, but hopefully productive, comments, are those who are currently blocked on Meta. I myself was blocked infinitely on Meta several years ago, but was allowed to contInue contributing on other wikimedia projects, such as this one. There are many long term contributors, some who have contrIbuted tens of thousands of edits to wikimedia projects who are are now blocked on specific projects. :Those who happen to be blocked on Meta have, to all intents and purposes, been removed from the wikimedia movement. :Ottawahitech [[Special:Contributions/&#126;2026-40514-81|&#126;2026-40514-81]] ([[User talk:&#126;2026-40514-81|talk]]) 21:41, 18 July 2026 (UTC) :*Hi Ottawahitech, thanks for the comment. Reviewing local bans on Meta is outside of my current scope, but if you have comments that you'd like to share here in the Colloquium relating to Wikiversity, I can consider the comments in the scope of the Café discussions. <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 20:05, 25 July 2026 (UTC) == Request for comment (the future of Abstract Wikipedia) == <bdi lang="en" dir="ltr" class="mw-content-ltr"> You are invited to voice your opinions in a [[:m:Requests for comment/The future of Abstract Wikipedia|request for comment about the future of Abstract Wikipedia]]. {{Int:Feedback-thanks-title}} [[:m:User:Kowal2701|Kowal2701]] ([[:m:User talk:Kowal2701|talk]]) 12:25, 24 July 2026 (UTC) </bdi> <!-- Message sent by User:DreamRimmer@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> 26r4mw6j6h7ri1jxhrbhknmxwqy7ydv 4 1791 2819712 2818442 2026-07-28T20:16:36Z Codename Noreste 2969951 /* Classical guitar pedagogy */ archive to [[Wikiversity:Requests for Deletion/Archives/24#Classical guitar pedagogy]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2819712 wikitext text/x-wiki {{/header}} == [[Concomitant Strabismus]] == {{archive top|Deleted per consensus. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:53, 16 July 2026 (UTC)}} Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC) :'''Delete''' per above. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:22, 12 July 2026 (UTC) {{archive bottom}} lzfkkkn5pszaoplll0w0hhdsabbyfom 2819714 2819712 2026-07-28T20:16:53Z Codename Noreste 2969951 /* Concomitant Strabismus */ archive to [[Wikiversity:Requests for Deletion/Archives/24#Concomitant Strabismus]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2819714 wikitext text/x-wiki {{/header}} rkzrjhaen6b84kyd39jlo1tw8y7f688 100 4719 2819678 2819098 2026-07-28T15:16:25Z Atcovi 276019 /* Projects and resources */ update 2819678 wikitext text/x-wiki {{center top}}<big>Welcome to the '''School of Environmental Sciences'''</big>, part of [[Portal:Interdisciplinary Studies|Interdisciplinary Studies]].{{center bottom}} __NOTOC__ A school is a large organizational structure which can contain various departments and divisions. The departments and divisions should be listed in the departments and divisions section. The school should not contain any learning resources. The school can contain [[Portal:Learning Projects|projects]] for developing [[Portal:Learning Materials|learning resources]]. ==Divisions and departments== Divisions and Departments of the School exist on pages in "topic" namespace. Start the name of departments with the "Topic:" prefix; departments reside in the [[Wikiversity:Namespaces|Topic: namespace]]. Departments and divisions link to learning materials and learning projects. Divisions can link subdivisions or to departments. For more information on schools, divisions and departments look at the [[Wikiversity:Naming_conventions|Naming Conventions]]. * [[Draft:Atmospheric sciences|Atmospheric sciences]] * [[Ecology]] * [[Exploration geology|Geosciences]] * [[Agronomy|Soil science]] * [[Hydrology]] * [[Environmental bioethics]] ==Projects and resources== * [[Earthinators: Plastic Pollution to Circular Solutions]] - A 5-week learning project on plastic pollution, policy, and circular solutions. *[[Permaculture]] - since 23 June 2016‎! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 05:54, 13 April 2018 (UTC) *[[Topic:Ecovillage|The Ecovillage Institute]] *[[The Spring Creek Project]] *[[Philippine Environmental Studies]] *[[AP Environmental Science]] ==Active participants== * Earthinators * Laird Steven H.Smith The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed). * Since 19 March 2016‎ with [[Agronomy]]! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 05:53, 13 April 2018 (UTC) ===Inactive participants=== *Stilite ==Things you can do!== * Clean up [[Draft:Geochemistry]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Geology]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Hydromorphology]] and move it to become a subpage of a supporting main page learning project. ==School news== * '''3 September 2006 ''' - School founded! == Thesis, papers and dissertations == * [[Scientific information on environmental issues in the age of digital media]] ==Collaborators== * [[User:Alarico|Alarico]] 21:07, 3 September 2006 (UTC) ==Wikimedia== * [[w: Environmental science]] ==See also== * [http://en.wikiversity.org/w/index.php?title=Special%3ASearch&ns0=1&ns4=1&ns100=1&ns102=1&ns104=1&search=Environmental_science&fulltext=Advanced+search Search Wikiversity for Environmental science ] [[Category:Environmental sciences| ]] [[Category:Wikiversity schools]] [[it:Corso:Scienze Ambientali]] bgry371xsur3qk7h3tfpvc4xc6y6qq6 3 53026 2819660 2818767 2026-07-28T14:55:05Z Tet-Math3 3062596 /* Please change the name for me. */ Reply 2819660 wikitext text/x-wiki <!-- {{Out of town}} --> <!-- {{Long wikibreak|image=Leaf_1_web.jpg|[[User:Jtneill|Jtneill]]|mid-Jan, 2012.}} --> {{{{TALKPAGENAME}}/Header}} {{TOCright}} == Moving to personal namespace == What are the policies or customs on Wikiversity for moving pages to personal userspace? Isn't there a risk that Wikiversity will turn into a blogging platform where many users will cultivate pages in their userspace and the outside world will not benefit from it? I see moving to ns user as a frequent suggestion in Requests for deletion (RFD). I would understand moving to ns Draft, which is clearly defined and there is a chance that the resource will then get into the main ns, thus serving the community. I would understand the suggestion to move to another wikiproject, where the text will serve the community. But I don't really understand the frequent moves to personal ns. Since it's in the RFD, it should either be kept or deleted. If someone contributes to Wikiversity, they automatically agree to its policies and also to the fact that they don't own the pages and someone can put them up for deletion. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:36, 22 November 2025 (UTC) I personally don't need a free website to host my pages. How would I get rid of the unfinished [[Pomology]] meta course if it was moved to my NS? ([https://en.wikiversity.org/wiki/Wikiversity:Requests_for_Deletion#c-Dan_Polansky-20251121091100-Juandev-20251120220900 Moving it to my own NS is suggested in RFD]). I'm putting it in the Request for deletion because, even though I started it, it looks like other editors had significant input there. Will I have the right to request speedy deletion if the pages are moved to my user ns? I think this tactic of moving to personal space is poorly thought out, but it has become the norm. Is there any guideline or discussion from before? If something appears in a deletion request, the majority decides that it should be moved to user ns, how can the person in question defend themselves that they don't want it in their own ns? It seems the community is pressuring the original author to agree to deletion. It seems that the user ns is an untouchable territory into which the community has the right to throw whatever it thinks from the main ns. So why aren't those pages deleted when the community decides that they don't belong in the main ns? --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:30, 22 November 2025 (UTC) {{ping|Juandev}} I replied on your talk page. But here's another version: Personally, in general, I try to keep my notes etc. in user space. Then if I have something more developed to share and collaborate on, then main space. Draft could be helpful to keep main space tidy, but is very quiet/unused, so in reality most drafts are in main space. But if the content is dubious, underdeveloped, lacking citation/peer review etc. then delete, or user space if it could still be developed. That's roughly how I see it. But everyone has a slightly different view/preference, so discuss to develop consensus. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:48, 22 November 2025 (UTC) == AI slop, ownership, and wikilawyering. == Using AI images is worse than no images. Your constant reverting of reasonable edits removing images you prompted on pages you wrote would be considered [[w:wp:OWN]]ership on Wikipedia; even if there is no general guideline on Wikiversity the spirit of not having the final say because just you made the page is applicable to all Wikimedia wikis. Reverting a reasonable edit because it lacks an image seems like [[w:wp:WIKILAWYER]]ing— I don’t know if edit summaries are ''required'' here, but I doubt it, and on most wikis they are simply recommended. Not having one doesn’t invalidate the edit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:27, 26 April 2026 (UTC) :I understand that you don't like many AI images because you consider them slop. My view is that some of these AI images can be useful for educational purposes. :I understand that you think an alternative or no image is better than some AI images. My view is that some AI images are better than no image and are either useful in addition to alternative images or more useful than some alternatives. :May I suggest deciding first on Commons whether to keep an image, rather than removing from Wikiversity and then nominating for deletion on Commons because of no use. :I have no interest in edit warring. I'll invite [[WV:RCA]] to review your recent edits. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:20, 26 April 2026 (UTC) == Hi. Would it be ok to post on your talk page using "AI"/LLMs? == Hello! Would it be ok if I posted some future messages that were generated by an "AI"/AI/LLM? If yes, would you prefer the generated message to be ie. max 100 words, less words or the talk message to include both original and generated message? Any other preferences/requirements? So far, 1 user has responded to this type of inquiry. They prefer 100 words max of generated talk page message. Best wishes [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:04, 22 June 2026 (UTC) : You are welcome to post directly to my talk page if you think that is a good place for a conversation. Personally, I don't much care whether or not content is AI-generated, but note the principles suggested by [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 23 June 2026 (UTC) == Resources suitable for the main namespace == I noticed that we have a Draft namespace on Wikiversity and that there were discussions about moving pages to Draft. Some colleagues also hold the opinion that pages should be moved to the user ns. However, I did not understand what the criteria are for such transfers, in other words, what page deserves to be on Wikiversity, but cannot be in the main ns. Now that I have [[:cs:Wikiverzita:Diskusní prostor#Ukončení činnosti na projektuh Wikiversity|finally left the Czech Wikiversity]], I am wondering if it is worth cloning my resources to the English one, or continuing on a personal wiki. For example, due to the resistance against AI-generated files on Commons, which has also spilled over to en.wv, I decided that I would not continue with [[Audio-visual German language materials]], because I wanted to generate the missing recordings and files in AI. This means that I will finish this course on my PC, rather than falling into eternal conjectures about why the AI ​​illustration of cherries is bad or good. And I have a similar concern with my other creations, where there was already pressure about a year ago to move them to a personal ns. What I have been creating in recent years has been education/learning through research. A person interested in a given topic asks a question and then researches the literature, or experiments and writes down the answer. Another person interested does the same, or as part of the training, looks for answers to other people's questions. The system may resemble Stack Overflow and the like, but the goal is not to create full texts together, but to go through the process of searching for information and learning from that. Of course, if the page is then too long, it can be turned into full-text study material and, for example, a new page of a similar nature can be founded. An example of such a project is needed [[Sweet Home 3D|here]] or [[User:Juandev/R/Compression stocking|here]], but there was an arumentation, they are underdeveloped and they should be moved to user ns. So that's why I'm asking what the evaluation criteria are, so that it doesn't end up in a way that the pages are moved away from the main ns and I end up finding out that I have to move it to my own wiki anyway. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:37, 3 July 2026 (UTC) : Oh, sorry to hear that you are finished with Czech Wikiversity, but maybe that is good for en.wv. : I guess we'll never really have any guarantees about anything placed on a publicly editable wiki because practices and users can change. : I share your concerns about actual, or threats of, rather blunt approaches to educational use of AI. Of course, AI can be educational useful, and of course we are capable of finding nuanced, reasonable ways to include and use it. But as we see e.g., on Commons, there is a strong, simplistic anti-AI sentiment within the Wikimedia community. : I don't recall much discussion about, or use of the Draft ns on en.wv. I think it was probably created very early on, to replicate Wikipedia, where a draft article makes sense before being moved to main space. I think the Draft or User space is welcome to be used for almost anything within scope, without much tension or debate. Then there is the issue around what some users consider acceptable or not for the main space on en.wv. Personally, I'm quite open. en.wv is still in early days of experimentation and trying things is needed, so I'm included to be inclusive and accepting, rather than shunting projects into Draft ns. : I don't use Draft:, but I do use User: subpages and of course main space. I haven't had any issues with others asking me to justify main space content or proposals to move content to Draft or User. : I'm sorry this doesn't provide any guarantees, except I guess to say I feel good about using en.wv as a working environment. : Sincerely,<br> James : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:25, 3 July 2026 (UTC) ::Well, yeah. I was just wondering if there was a debate around Draft, but if werent its about the opinion of the future community. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:51, 17 July 2026 (UTC) ::: I don't recall general, historical debate (but I wasn't around when en.wv was created). I am only aware of the more recent discussions in recent years (which I think you're aware of) about what to do with specific content propsoed for deletion or moving to user or draft space. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:43, 21 July 2026 (UTC) == LLM-generated content: Talk post about my neurodiversity draft == Hello Jtneill — does the topic I’ve brought up interest you? If so, would you be willing to spend about 5 minutes skimming through my “idea” on Wikiversity (“[[Draft:The Neurodiversity-inspired Idea]]”)? No need to overthink it, and I’m not expecting a reply soon—tomorrow, in a month, or even later would all be welcome. Thank you. Metadata: Since edit summaries are not present on talk pages like this one, I'll link to the LLM interaction history that I saved in my Wikiversity user space here: [[User:ThinkingScience/All_General_AI_Prompt_History_Archive#Goal:_Interact_with_User:Jtneill_at_July_5,_2026]] so that I follow [[Wikiversity:Artificial intelligence]]. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 04:38, 5 July 2026 (UTC) : Apologies, I've been on leave. I took a 5-second look. My first thought is: Can you name the idea? i.e., rather than calling the page "The Neurodiversity-inspired Idea", which doesn't really say very much, develop a name that describes the idea itself in half a dozen words. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:41, 21 July 2026 (UTC) == Nodal Current Splitter == Hello Jtneill ( I have been E-Mailed by you a few times in the past ) I had placed a page called Nodal Current Splitter on Wikiversity & it was a way to solve unbalanced bridge circuits using Ohms Law rather then having to use Kirchhoff's mesh equation method. It greatly reduced the amount of work that was required to solve such problems. It once showed up on your site when searching Wikiversity as Nodal Current Splitter. Again it allowed you to use Ohms Law instead of the harder method laid out by Kirchhoff. Can you tell me why the page was removed please ? I have another page that's still on Wikiversity called Trirectangular Tetrahedrons & I use the user name Tet-Math, 1 through 5 because Wikiversity prevents me from simply using Tet-Math more than once. Thank You. lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:21, 20 July 2026 (UTC) == Please change the name for me. == Hello again Jtneill I just thought of something I wished I had done in the 1st place. I think the Page Name should have been "Nodal Current Divider" all along because nobody searches for "Splitter" although that's what it actually does. Thank You again lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:37, 20 July 2026 (UTC) : Hello. Apologies, I may have lost the context. But you can "Move" the page yourself. Go to the page, and click "Move" (top right). Let me know if I can do anything else. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 21 July 2026 (UTC) ::Hello Jtnwill ::It's not that I'm trying to move my page - It's that it has ceased to exist. ::When I look for it here: https://en.wikiversity.org/w/index.php?search=Nodal+Current+Splitter&title=Special:Search&ns0=1 ::I get this: Showing results for [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&search=normal+current+splinter&fulltext=1&profile=default&wprov=srpw1_dymr1 normal current splinter]. No results found for Nodal Current Splitter. ::Create the page "[[Nodal Current Splitter]]" on this wiki! See also the search results found. ::Somebody has deleted a new method of solving complicated electronic problem & with it several new formulas !!! ::lm@[[User:Start.ca|Start.ca]] [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:24, 21 July 2026 (UTC) ::: Oh, I see. ::: Check your contributions here: [[Special:Contributions/Tet-Math4]] ::: You can also see if anything was deleted: [[Special:DeletedContributions/Tet-Math4]] ::: Also, maybe check if you contributed using another account or maybe when not logged in? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:30, 21 July 2026 (UTC) ::::Thanks Jtneill ::::I hit the [[Special:DeletedContributions/Tet-Math4]] but I just ran into this instead: ::::Permission error ::::You do not have permission to view metadata of deleted history entries, for the following reason: ::::The action you have requested is limited to users in the group: [[Wikiversity:Custodianship|Custodians]]. ::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:41, 21 July 2026 (UTC) ::::: OK, I checked - it is empty for this account (i.e., no deletions). -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:45, 21 July 2026 (UTC) ::::::Thanks Jtneill ::::::I have no idea why it's been deleted without a trace when it was on Wikiversity for many months. As a matter of fact it was the very 1st item showing when I typed in Nodal Current Splitter. ::::::But I will install it again & rename it Current Divider (Nodal). I can find all of the parts & pieces somewhere !!! ::::::Thanks Again ::::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 08:00, 21 July 2026 (UTC) ::::::Hello Jtneil ::::::My mistake. I did't realize that when TeT-Math4 popped up all by itself it would send you to the wrong place. My Nodal Current Splitter Sandbox is still there but it's TeT-Math3 !!! ::::::Again i say somebody deleted my work called Nodal Current Splitter. Please check into this for me. ::::::lm@start.ca [[User:Tet-Math3|Tet-Math3]] ([[User talk:Tet-Math3|discuss]] • [[Special:Contributions/Tet-Math3|contribs]]) 14:55, 28 July 2026 (UTC) ::::In my Sandbox at [[User:Tet-Math/sandbox|Users:Tet-Math/sandbox]] there is some of my Nodal Current Splitter page nut not nearly as elaborate ! ::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:49, 21 July 2026 (UTC) 9p2x9fxvejrd3re73i1d8oz5v6qeg5w 2819666 2819660 2026-07-28T15:06:39Z Atcovi 276019 /* Please change the name for me. */ Reply 2819666 wikitext text/x-wiki <!-- {{Out of town}} --> <!-- {{Long wikibreak|image=Leaf_1_web.jpg|[[User:Jtneill|Jtneill]]|mid-Jan, 2012.}} --> {{{{TALKPAGENAME}}/Header}} {{TOCright}} == Moving to personal namespace == What are the policies or customs on Wikiversity for moving pages to personal userspace? Isn't there a risk that Wikiversity will turn into a blogging platform where many users will cultivate pages in their userspace and the outside world will not benefit from it? I see moving to ns user as a frequent suggestion in Requests for deletion (RFD). I would understand moving to ns Draft, which is clearly defined and there is a chance that the resource will then get into the main ns, thus serving the community. I would understand the suggestion to move to another wikiproject, where the text will serve the community. But I don't really understand the frequent moves to personal ns. Since it's in the RFD, it should either be kept or deleted. If someone contributes to Wikiversity, they automatically agree to its policies and also to the fact that they don't own the pages and someone can put them up for deletion. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:36, 22 November 2025 (UTC) I personally don't need a free website to host my pages. How would I get rid of the unfinished [[Pomology]] meta course if it was moved to my NS? ([https://en.wikiversity.org/wiki/Wikiversity:Requests_for_Deletion#c-Dan_Polansky-20251121091100-Juandev-20251120220900 Moving it to my own NS is suggested in RFD]). I'm putting it in the Request for deletion because, even though I started it, it looks like other editors had significant input there. Will I have the right to request speedy deletion if the pages are moved to my user ns? I think this tactic of moving to personal space is poorly thought out, but it has become the norm. Is there any guideline or discussion from before? If something appears in a deletion request, the majority decides that it should be moved to user ns, how can the person in question defend themselves that they don't want it in their own ns? It seems the community is pressuring the original author to agree to deletion. It seems that the user ns is an untouchable territory into which the community has the right to throw whatever it thinks from the main ns. So why aren't those pages deleted when the community decides that they don't belong in the main ns? --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:30, 22 November 2025 (UTC) {{ping|Juandev}} I replied on your talk page. But here's another version: Personally, in general, I try to keep my notes etc. in user space. Then if I have something more developed to share and collaborate on, then main space. Draft could be helpful to keep main space tidy, but is very quiet/unused, so in reality most drafts are in main space. But if the content is dubious, underdeveloped, lacking citation/peer review etc. then delete, or user space if it could still be developed. That's roughly how I see it. But everyone has a slightly different view/preference, so discuss to develop consensus. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:48, 22 November 2025 (UTC) == AI slop, ownership, and wikilawyering. == Using AI images is worse than no images. Your constant reverting of reasonable edits removing images you prompted on pages you wrote would be considered [[w:wp:OWN]]ership on Wikipedia; even if there is no general guideline on Wikiversity the spirit of not having the final say because just you made the page is applicable to all Wikimedia wikis. Reverting a reasonable edit because it lacks an image seems like [[w:wp:WIKILAWYER]]ing— I don’t know if edit summaries are ''required'' here, but I doubt it, and on most wikis they are simply recommended. Not having one doesn’t invalidate the edit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:27, 26 April 2026 (UTC) :I understand that you don't like many AI images because you consider them slop. My view is that some of these AI images can be useful for educational purposes. :I understand that you think an alternative or no image is better than some AI images. My view is that some AI images are better than no image and are either useful in addition to alternative images or more useful than some alternatives. :May I suggest deciding first on Commons whether to keep an image, rather than removing from Wikiversity and then nominating for deletion on Commons because of no use. :I have no interest in edit warring. I'll invite [[WV:RCA]] to review your recent edits. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:20, 26 April 2026 (UTC) == Hi. Would it be ok to post on your talk page using "AI"/LLMs? == Hello! Would it be ok if I posted some future messages that were generated by an "AI"/AI/LLM? If yes, would you prefer the generated message to be ie. max 100 words, less words or the talk message to include both original and generated message? Any other preferences/requirements? So far, 1 user has responded to this type of inquiry. They prefer 100 words max of generated talk page message. Best wishes [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:04, 22 June 2026 (UTC) : You are welcome to post directly to my talk page if you think that is a good place for a conversation. Personally, I don't much care whether or not content is AI-generated, but note the principles suggested by [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 23 June 2026 (UTC) == Resources suitable for the main namespace == I noticed that we have a Draft namespace on Wikiversity and that there were discussions about moving pages to Draft. Some colleagues also hold the opinion that pages should be moved to the user ns. However, I did not understand what the criteria are for such transfers, in other words, what page deserves to be on Wikiversity, but cannot be in the main ns. Now that I have [[:cs:Wikiverzita:Diskusní prostor#Ukončení činnosti na projektuh Wikiversity|finally left the Czech Wikiversity]], I am wondering if it is worth cloning my resources to the English one, or continuing on a personal wiki. For example, due to the resistance against AI-generated files on Commons, which has also spilled over to en.wv, I decided that I would not continue with [[Audio-visual German language materials]], because I wanted to generate the missing recordings and files in AI. This means that I will finish this course on my PC, rather than falling into eternal conjectures about why the AI ​​illustration of cherries is bad or good. And I have a similar concern with my other creations, where there was already pressure about a year ago to move them to a personal ns. What I have been creating in recent years has been education/learning through research. A person interested in a given topic asks a question and then researches the literature, or experiments and writes down the answer. Another person interested does the same, or as part of the training, looks for answers to other people's questions. The system may resemble Stack Overflow and the like, but the goal is not to create full texts together, but to go through the process of searching for information and learning from that. Of course, if the page is then too long, it can be turned into full-text study material and, for example, a new page of a similar nature can be founded. An example of such a project is needed [[Sweet Home 3D|here]] or [[User:Juandev/R/Compression stocking|here]], but there was an arumentation, they are underdeveloped and they should be moved to user ns. So that's why I'm asking what the evaluation criteria are, so that it doesn't end up in a way that the pages are moved away from the main ns and I end up finding out that I have to move it to my own wiki anyway. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:37, 3 July 2026 (UTC) : Oh, sorry to hear that you are finished with Czech Wikiversity, but maybe that is good for en.wv. : I guess we'll never really have any guarantees about anything placed on a publicly editable wiki because practices and users can change. : I share your concerns about actual, or threats of, rather blunt approaches to educational use of AI. Of course, AI can be educational useful, and of course we are capable of finding nuanced, reasonable ways to include and use it. But as we see e.g., on Commons, there is a strong, simplistic anti-AI sentiment within the Wikimedia community. : I don't recall much discussion about, or use of the Draft ns on en.wv. I think it was probably created very early on, to replicate Wikipedia, where a draft article makes sense before being moved to main space. I think the Draft or User space is welcome to be used for almost anything within scope, without much tension or debate. Then there is the issue around what some users consider acceptable or not for the main space on en.wv. Personally, I'm quite open. en.wv is still in early days of experimentation and trying things is needed, so I'm included to be inclusive and accepting, rather than shunting projects into Draft ns. : I don't use Draft:, but I do use User: subpages and of course main space. I haven't had any issues with others asking me to justify main space content or proposals to move content to Draft or User. : I'm sorry this doesn't provide any guarantees, except I guess to say I feel good about using en.wv as a working environment. : Sincerely,<br> James : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:25, 3 July 2026 (UTC) ::Well, yeah. I was just wondering if there was a debate around Draft, but if werent its about the opinion of the future community. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:51, 17 July 2026 (UTC) ::: I don't recall general, historical debate (but I wasn't around when en.wv was created). I am only aware of the more recent discussions in recent years (which I think you're aware of) about what to do with specific content propsoed for deletion or moving to user or draft space. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:43, 21 July 2026 (UTC) == LLM-generated content: Talk post about my neurodiversity draft == Hello Jtneill — does the topic I’ve brought up interest you? If so, would you be willing to spend about 5 minutes skimming through my “idea” on Wikiversity (“[[Draft:The Neurodiversity-inspired Idea]]”)? No need to overthink it, and I’m not expecting a reply soon—tomorrow, in a month, or even later would all be welcome. Thank you. Metadata: Since edit summaries are not present on talk pages like this one, I'll link to the LLM interaction history that I saved in my Wikiversity user space here: [[User:ThinkingScience/All_General_AI_Prompt_History_Archive#Goal:_Interact_with_User:Jtneill_at_July_5,_2026]] so that I follow [[Wikiversity:Artificial intelligence]]. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 04:38, 5 July 2026 (UTC) : Apologies, I've been on leave. I took a 5-second look. My first thought is: Can you name the idea? i.e., rather than calling the page "The Neurodiversity-inspired Idea", which doesn't really say very much, develop a name that describes the idea itself in half a dozen words. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:41, 21 July 2026 (UTC) == Nodal Current Splitter == Hello Jtneill ( I have been E-Mailed by you a few times in the past ) I had placed a page called Nodal Current Splitter on Wikiversity & it was a way to solve unbalanced bridge circuits using Ohms Law rather then having to use Kirchhoff's mesh equation method. It greatly reduced the amount of work that was required to solve such problems. It once showed up on your site when searching Wikiversity as Nodal Current Splitter. Again it allowed you to use Ohms Law instead of the harder method laid out by Kirchhoff. Can you tell me why the page was removed please ? I have another page that's still on Wikiversity called Trirectangular Tetrahedrons & I use the user name Tet-Math, 1 through 5 because Wikiversity prevents me from simply using Tet-Math more than once. Thank You. lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:21, 20 July 2026 (UTC) == Please change the name for me. == Hello again Jtneill I just thought of something I wished I had done in the 1st place. I think the Page Name should have been "Nodal Current Divider" all along because nobody searches for "Splitter" although that's what it actually does. Thank You again lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:37, 20 July 2026 (UTC) : Hello. Apologies, I may have lost the context. But you can "Move" the page yourself. Go to the page, and click "Move" (top right). Let me know if I can do anything else. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 21 July 2026 (UTC) ::Hello Jtnwill ::It's not that I'm trying to move my page - It's that it has ceased to exist. ::When I look for it here: https://en.wikiversity.org/w/index.php?search=Nodal+Current+Splitter&title=Special:Search&ns0=1 ::I get this: Showing results for [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&search=normal+current+splinter&fulltext=1&profile=default&wprov=srpw1_dymr1 normal current splinter]. No results found for Nodal Current Splitter. ::Create the page "[[Nodal Current Splitter]]" on this wiki! See also the search results found. ::Somebody has deleted a new method of solving complicated electronic problem & with it several new formulas !!! ::lm@[[User:Start.ca|Start.ca]] [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:24, 21 July 2026 (UTC) ::: Oh, I see. ::: Check your contributions here: [[Special:Contributions/Tet-Math4]] ::: You can also see if anything was deleted: [[Special:DeletedContributions/Tet-Math4]] ::: Also, maybe check if you contributed using another account or maybe when not logged in? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:30, 21 July 2026 (UTC) ::::Thanks Jtneill ::::I hit the [[Special:DeletedContributions/Tet-Math4]] but I just ran into this instead: ::::Permission error ::::You do not have permission to view metadata of deleted history entries, for the following reason: ::::The action you have requested is limited to users in the group: [[Wikiversity:Custodianship|Custodians]]. ::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:41, 21 July 2026 (UTC) ::::: OK, I checked - it is empty for this account (i.e., no deletions). -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:45, 21 July 2026 (UTC) ::::::Thanks Jtneill ::::::I have no idea why it's been deleted without a trace when it was on Wikiversity for many months. As a matter of fact it was the very 1st item showing when I typed in Nodal Current Splitter. ::::::But I will install it again & rename it Current Divider (Nodal). I can find all of the parts & pieces somewhere !!! ::::::Thanks Again ::::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 08:00, 21 July 2026 (UTC) ::::::Hello Jtneil ::::::My mistake. I did't realize that when TeT-Math4 popped up all by itself it would send you to the wrong place. My Nodal Current Splitter Sandbox is still there but it's TeT-Math3 !!! ::::::Again i say somebody deleted my work called Nodal Current Splitter. Please check into this for me. ::::::lm@start.ca [[User:Tet-Math3|Tet-Math3]] ([[User talk:Tet-Math3|discuss]] • [[Special:Contributions/Tet-Math3|contribs]]) 14:55, 28 July 2026 (UTC) ::::In my Sandbox at [[User:Tet-Math/sandbox|Users:Tet-Math/sandbox]] there is some of my Nodal Current Splitter page nut not nearly as elaborate ! ::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:49, 21 July 2026 (UTC) :::::The pages you created were deleted as they were not within Wikiversity's [[Wikiversity:Research ethics|research ethics]] (the specific reasoning is as so: "No [[Wikiversity:WIW|educational objectives or discussion in history]]: classic pseudoscience: inflated language, lack of citations, lack of empirical validation"). Please don't recreate these pages in the mainspace, and using multiple accounts to post the same disruptive content in inappropriate spaces may result in a ban. BTW, [[User:Tet-Math3/Trirectangular Tetrahedrons]] was also moved into your userspace for it being low-quality. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:06, 28 July 2026 (UTC) lo4qd0pj19k01ev1l0fqt8n6kkttjky 2819746 2819666 2026-07-29T05:58:14Z Tet-Math3 3062596 /* Please change the name for me. */ Reply 2819746 wikitext text/x-wiki <!-- {{Out of town}} --> <!-- {{Long wikibreak|image=Leaf_1_web.jpg|[[User:Jtneill|Jtneill]]|mid-Jan, 2012.}} --> {{{{TALKPAGENAME}}/Header}} {{TOCright}} == Moving to personal namespace == What are the policies or customs on Wikiversity for moving pages to personal userspace? Isn't there a risk that Wikiversity will turn into a blogging platform where many users will cultivate pages in their userspace and the outside world will not benefit from it? I see moving to ns user as a frequent suggestion in Requests for deletion (RFD). I would understand moving to ns Draft, which is clearly defined and there is a chance that the resource will then get into the main ns, thus serving the community. I would understand the suggestion to move to another wikiproject, where the text will serve the community. But I don't really understand the frequent moves to personal ns. Since it's in the RFD, it should either be kept or deleted. If someone contributes to Wikiversity, they automatically agree to its policies and also to the fact that they don't own the pages and someone can put them up for deletion. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:36, 22 November 2025 (UTC) I personally don't need a free website to host my pages. How would I get rid of the unfinished [[Pomology]] meta course if it was moved to my NS? ([https://en.wikiversity.org/wiki/Wikiversity:Requests_for_Deletion#c-Dan_Polansky-20251121091100-Juandev-20251120220900 Moving it to my own NS is suggested in RFD]). I'm putting it in the Request for deletion because, even though I started it, it looks like other editors had significant input there. Will I have the right to request speedy deletion if the pages are moved to my user ns? I think this tactic of moving to personal space is poorly thought out, but it has become the norm. Is there any guideline or discussion from before? If something appears in a deletion request, the majority decides that it should be moved to user ns, how can the person in question defend themselves that they don't want it in their own ns? It seems the community is pressuring the original author to agree to deletion. It seems that the user ns is an untouchable territory into which the community has the right to throw whatever it thinks from the main ns. So why aren't those pages deleted when the community decides that they don't belong in the main ns? --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:30, 22 November 2025 (UTC) {{ping|Juandev}} I replied on your talk page. But here's another version: Personally, in general, I try to keep my notes etc. in user space. Then if I have something more developed to share and collaborate on, then main space. Draft could be helpful to keep main space tidy, but is very quiet/unused, so in reality most drafts are in main space. But if the content is dubious, underdeveloped, lacking citation/peer review etc. then delete, or user space if it could still be developed. That's roughly how I see it. But everyone has a slightly different view/preference, so discuss to develop consensus. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:48, 22 November 2025 (UTC) == AI slop, ownership, and wikilawyering. == Using AI images is worse than no images. Your constant reverting of reasonable edits removing images you prompted on pages you wrote would be considered [[w:wp:OWN]]ership on Wikipedia; even if there is no general guideline on Wikiversity the spirit of not having the final say because just you made the page is applicable to all Wikimedia wikis. Reverting a reasonable edit because it lacks an image seems like [[w:wp:WIKILAWYER]]ing— I don’t know if edit summaries are ''required'' here, but I doubt it, and on most wikis they are simply recommended. Not having one doesn’t invalidate the edit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:27, 26 April 2026 (UTC) :I understand that you don't like many AI images because you consider them slop. My view is that some of these AI images can be useful for educational purposes. :I understand that you think an alternative or no image is better than some AI images. My view is that some AI images are better than no image and are either useful in addition to alternative images or more useful than some alternatives. :May I suggest deciding first on Commons whether to keep an image, rather than removing from Wikiversity and then nominating for deletion on Commons because of no use. :I have no interest in edit warring. I'll invite [[WV:RCA]] to review your recent edits. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:20, 26 April 2026 (UTC) == Hi. Would it be ok to post on your talk page using "AI"/LLMs? == Hello! Would it be ok if I posted some future messages that were generated by an "AI"/AI/LLM? If yes, would you prefer the generated message to be ie. max 100 words, less words or the talk message to include both original and generated message? Any other preferences/requirements? So far, 1 user has responded to this type of inquiry. They prefer 100 words max of generated talk page message. Best wishes [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:04, 22 June 2026 (UTC) : You are welcome to post directly to my talk page if you think that is a good place for a conversation. Personally, I don't much care whether or not content is AI-generated, but note the principles suggested by [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 23 June 2026 (UTC) == Resources suitable for the main namespace == I noticed that we have a Draft namespace on Wikiversity and that there were discussions about moving pages to Draft. Some colleagues also hold the opinion that pages should be moved to the user ns. However, I did not understand what the criteria are for such transfers, in other words, what page deserves to be on Wikiversity, but cannot be in the main ns. Now that I have [[:cs:Wikiverzita:Diskusní prostor#Ukončení činnosti na projektuh Wikiversity|finally left the Czech Wikiversity]], I am wondering if it is worth cloning my resources to the English one, or continuing on a personal wiki. For example, due to the resistance against AI-generated files on Commons, which has also spilled over to en.wv, I decided that I would not continue with [[Audio-visual German language materials]], because I wanted to generate the missing recordings and files in AI. This means that I will finish this course on my PC, rather than falling into eternal conjectures about why the AI ​​illustration of cherries is bad or good. And I have a similar concern with my other creations, where there was already pressure about a year ago to move them to a personal ns. What I have been creating in recent years has been education/learning through research. A person interested in a given topic asks a question and then researches the literature, or experiments and writes down the answer. Another person interested does the same, or as part of the training, looks for answers to other people's questions. The system may resemble Stack Overflow and the like, but the goal is not to create full texts together, but to go through the process of searching for information and learning from that. Of course, if the page is then too long, it can be turned into full-text study material and, for example, a new page of a similar nature can be founded. An example of such a project is needed [[Sweet Home 3D|here]] or [[User:Juandev/R/Compression stocking|here]], but there was an arumentation, they are underdeveloped and they should be moved to user ns. So that's why I'm asking what the evaluation criteria are, so that it doesn't end up in a way that the pages are moved away from the main ns and I end up finding out that I have to move it to my own wiki anyway. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:37, 3 July 2026 (UTC) : Oh, sorry to hear that you are finished with Czech Wikiversity, but maybe that is good for en.wv. : I guess we'll never really have any guarantees about anything placed on a publicly editable wiki because practices and users can change. : I share your concerns about actual, or threats of, rather blunt approaches to educational use of AI. Of course, AI can be educational useful, and of course we are capable of finding nuanced, reasonable ways to include and use it. But as we see e.g., on Commons, there is a strong, simplistic anti-AI sentiment within the Wikimedia community. : I don't recall much discussion about, or use of the Draft ns on en.wv. I think it was probably created very early on, to replicate Wikipedia, where a draft article makes sense before being moved to main space. I think the Draft or User space is welcome to be used for almost anything within scope, without much tension or debate. Then there is the issue around what some users consider acceptable or not for the main space on en.wv. Personally, I'm quite open. en.wv is still in early days of experimentation and trying things is needed, so I'm included to be inclusive and accepting, rather than shunting projects into Draft ns. : I don't use Draft:, but I do use User: subpages and of course main space. I haven't had any issues with others asking me to justify main space content or proposals to move content to Draft or User. : I'm sorry this doesn't provide any guarantees, except I guess to say I feel good about using en.wv as a working environment. : Sincerely,<br> James : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:25, 3 July 2026 (UTC) ::Well, yeah. I was just wondering if there was a debate around Draft, but if werent its about the opinion of the future community. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:51, 17 July 2026 (UTC) ::: I don't recall general, historical debate (but I wasn't around when en.wv was created). I am only aware of the more recent discussions in recent years (which I think you're aware of) about what to do with specific content propsoed for deletion or moving to user or draft space. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:43, 21 July 2026 (UTC) == LLM-generated content: Talk post about my neurodiversity draft == Hello Jtneill — does the topic I’ve brought up interest you? If so, would you be willing to spend about 5 minutes skimming through my “idea” on Wikiversity (“[[Draft:The Neurodiversity-inspired Idea]]”)? No need to overthink it, and I’m not expecting a reply soon—tomorrow, in a month, or even later would all be welcome. Thank you. Metadata: Since edit summaries are not present on talk pages like this one, I'll link to the LLM interaction history that I saved in my Wikiversity user space here: [[User:ThinkingScience/All_General_AI_Prompt_History_Archive#Goal:_Interact_with_User:Jtneill_at_July_5,_2026]] so that I follow [[Wikiversity:Artificial intelligence]]. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 04:38, 5 July 2026 (UTC) : Apologies, I've been on leave. I took a 5-second look. My first thought is: Can you name the idea? i.e., rather than calling the page "The Neurodiversity-inspired Idea", which doesn't really say very much, develop a name that describes the idea itself in half a dozen words. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:41, 21 July 2026 (UTC) == Nodal Current Splitter == Hello Jtneill ( I have been E-Mailed by you a few times in the past ) I had placed a page called Nodal Current Splitter on Wikiversity & it was a way to solve unbalanced bridge circuits using Ohms Law rather then having to use Kirchhoff's mesh equation method. It greatly reduced the amount of work that was required to solve such problems. It once showed up on your site when searching Wikiversity as Nodal Current Splitter. Again it allowed you to use Ohms Law instead of the harder method laid out by Kirchhoff. Can you tell me why the page was removed please ? I have another page that's still on Wikiversity called Trirectangular Tetrahedrons & I use the user name Tet-Math, 1 through 5 because Wikiversity prevents me from simply using Tet-Math more than once. Thank You. lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:21, 20 July 2026 (UTC) == Please change the name for me. == Hello again Jtneill I just thought of something I wished I had done in the 1st place. I think the Page Name should have been "Nodal Current Divider" all along because nobody searches for "Splitter" although that's what it actually does. Thank You again lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 05:37, 20 July 2026 (UTC) : Hello. Apologies, I may have lost the context. But you can "Move" the page yourself. Go to the page, and click "Move" (top right). Let me know if I can do anything else. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:39, 21 July 2026 (UTC) ::Hello Jtnwill ::It's not that I'm trying to move my page - It's that it has ceased to exist. ::When I look for it here: https://en.wikiversity.org/w/index.php?search=Nodal+Current+Splitter&title=Special:Search&ns0=1 ::I get this: Showing results for [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&search=normal+current+splinter&fulltext=1&profile=default&wprov=srpw1_dymr1 normal current splinter]. No results found for Nodal Current Splitter. ::Create the page "[[Nodal Current Splitter]]" on this wiki! See also the search results found. ::Somebody has deleted a new method of solving complicated electronic problem & with it several new formulas !!! ::lm@[[User:Start.ca|Start.ca]] [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:24, 21 July 2026 (UTC) ::: Oh, I see. ::: Check your contributions here: [[Special:Contributions/Tet-Math4]] ::: You can also see if anything was deleted: [[Special:DeletedContributions/Tet-Math4]] ::: Also, maybe check if you contributed using another account or maybe when not logged in? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:30, 21 July 2026 (UTC) ::::Thanks Jtneill ::::I hit the [[Special:DeletedContributions/Tet-Math4]] but I just ran into this instead: ::::Permission error ::::You do not have permission to view metadata of deleted history entries, for the following reason: ::::The action you have requested is limited to users in the group: [[Wikiversity:Custodianship|Custodians]]. ::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:41, 21 July 2026 (UTC) ::::: OK, I checked - it is empty for this account (i.e., no deletions). -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 07:45, 21 July 2026 (UTC) ::::::Thanks Jtneill ::::::I have no idea why it's been deleted without a trace when it was on Wikiversity for many months. As a matter of fact it was the very 1st item showing when I typed in Nodal Current Splitter. ::::::But I will install it again & rename it Current Divider (Nodal). I can find all of the parts & pieces somewhere !!! ::::::Thanks Again ::::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 08:00, 21 July 2026 (UTC) ::::::Hello Jtneil ::::::My mistake. I did't realize that when TeT-Math4 popped up all by itself it would send you to the wrong place. My Nodal Current Splitter Sandbox is still there but it's TeT-Math3 !!! ::::::Again i say somebody deleted my work called Nodal Current Splitter. Please check into this for me. ::::::lm@start.ca [[User:Tet-Math3|Tet-Math3]] ([[User talk:Tet-Math3|discuss]] • [[Special:Contributions/Tet-Math3|contribs]]) 14:55, 28 July 2026 (UTC) ::::In my Sandbox at [[User:Tet-Math/sandbox|Users:Tet-Math/sandbox]] there is some of my Nodal Current Splitter page nut not nearly as elaborate ! ::::lm@start.ca [[User:Tet-Math4|Tet-Math4]] ([[User talk:Tet-Math4|discuss]] • [[Special:Contributions/Tet-Math4|contribs]]) 07:49, 21 July 2026 (UTC) :::::The pages you created were deleted as they were not within Wikiversity's [[Wikiversity:Research ethics|research ethics]] (the specific reasoning is as so: "No [[Wikiversity:WIW|educational objectives or discussion in history]]: classic pseudoscience: inflated language, lack of citations, lack of empirical validation"). Please don't recreate these pages in the mainspace, and using multiple accounts to post the same disruptive content in inappropriate spaces may result in a ban. BTW, [[User:Tet-Math3/Trirectangular Tetrahedrons]] was also moved into your userspace for it being low-quality. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:06, 28 July 2026 (UTC) ::::::Well it's too bad that you could not see the equations presented in both cases actually worked & are NEW !!! ::::::I however will not attempt to present any information on Wikiversity again !!! ::::::lm@start.ca [[User:Tet-Math3|Tet-Math3]] ([[User talk:Tet-Math3|discuss]] • [[Special:Contributions/Tet-Math3|contribs]]) 05:58, 29 July 2026 (UTC) tt9pjq5harx70c0ch0xqc5tkgedslha 0 111291 2819719 2263454 2026-07-28T23:54:51Z PieWriter 3039865 2819719 wikitext text/x-wiki == Edible Landscaping == Edible landscaping is the conversion of land spaces, especially marginal land spaces such as flowerbeds along a sidewalk, into working spaces growing food producing plant otherwise known as edible plants. This can be achieved through exercising techniques of permaculture where plants are selected to grow with complementary plants while respecting the immediate local environmental conditions. Using topiary and ornamental design with these edible plants also offers the opportunity to convert the otherwise ornamental botanic gardens into productive working spaces. That is, an edible landscape can use techniques of permaculture, which selects plants to grow with complementary plants while respecting the immediate local environmental conditions. Topiary and ornamental designs convert edible plants into ornamental botanic shapes that combine the concept of a botanical garden and food-productive working spaces. Edible landscaping is not a new idea; edible landscaping dates back well over two thousand years to the hanging gardens of Babylon. The Garden of Eden is also a biblical example. Edible landscaping was also an underlying principle to medieval monastic gardens. Australian naturist, [[Wikipedia:Bill Mollison|Bill Mollison]] created a renewed interest in edible landscaping in the 1970's, through his concept of [[Wikipedia:Permaculture|permaculture]]. Permaculture is a landscape design technique designed to form harmonious balances between input and outputs. == University Involvement == Universities as a center of higher learning have an obligatory role in the community to research and introduce edible landscapes. These edible landscapes are a matter of social justice The majority of food in this country travels 1500 miles to reach a consumer’s plate. The fossil fuels necessary to transport produce this absurd distance creates a massive carbon footprint that contributes to global warming. Additionally, this food is oftentimes grown with hazardous pesticides and herbicides, affecting the health of the farm workers. Edible landscapes can demonstrates that there are alternatives to industrial agriculture. The landscape will spur discussion of current farming practices and sustainable agricultural practices while providing students with food that is grown for the nourishment of body and spirit. ===Key benefits of an edible landscape in university=== * Edible landscapes can serve as a living landscape for academic exercises * Edible landscaping is a mater of social justice to allow access of nutritious food to all people. * Positive mental health. == List of Universities participating in edible landscaping projects. == * Loyola University Chicago - [[Loyola_University_Chicago_-_Project_Eden|Project Eden]] * University Of Minnesota - [http://umediblelandscape.blogspot.com/ UM Edible Landscape] * Scripps College - [http://sustainability.claremont.edu/food/index.html Food and Sustainability Community Project] [[Category:Architecture]] [[Category:Permaculture]] [[Category:Universities]] dsxxpofbv1tnxhs1ft2xfj01qh0tgvb 2819743 2819719 2026-07-29T02:44:27Z Jtneill 10242 2819743 wikitext text/x-wiki == Edible Landscaping == Edible landscaping is the conversion of land spaces, especially marginal land spaces such as flowerbeds along a sidewalk, into working spaces growing food producing plant otherwise known as edible plants. This can be achieved through exercising techniques of permaculture where plants are selected to grow with complementary plants while respecting the immediate local environmental conditions. Using topiary and ornamental design with these edible plants also offers the opportunity to convert the otherwise ornamental botanic gardens into productive working spaces. That is, an edible landscape can use techniques of permaculture, which selects plants to grow with complementary plants while respecting the immediate local environmental conditions. Topiary and ornamental designs convert edible plants into ornamental botanic shapes that combine the concept of a botanical garden and food-productive working spaces. Edible landscaping is not a new idea; edible landscaping dates back well over two thousand years to the hanging gardens of Babylon. The Garden of Eden is also a biblical example. Edible landscaping was also an underlying principle to medieval monastic gardens. Australian naturist, [[Wikipedia:Bill Mollison|Bill Mollison]] created a renewed interest in edible landscaping in the 1970's, through his concept of [[Wikipedia:Permaculture|permaculture]]. Permaculture is a landscape design technique designed to form harmonious balances between input and outputs. == University Involvement == Universities as a center of higher learning have an obligatory role in the community to research and introduce edible landscapes. These edible landscapes are a matter of social justice The majority of food in this country travels 1500 miles to reach a consumer’s plate. The fossil fuels necessary to transport produce this absurd distance creates a massive carbon footprint that contributes to global warming. Additionally, this food is oftentimes grown with hazardous pesticides and herbicides, affecting the health of the farm workers. Edible landscapes can demonstrates that there are alternatives to industrial agriculture. The landscape will spur discussion of current farming practices and sustainable agricultural practices while providing students with food that is grown for the nourishment of body and spirit. ===Key benefits of an edible landscape in university=== * Edible landscapes can serve as a living landscape for academic exercises * Edible landscaping is a mater of social justice to allow access of nutritious food to all people. * Positive mental health. == List of Universities participating in edible landscaping projects. == * Loyola University Chicago - [[Loyola_University_Chicago_-_Project_Eden|Project Eden]] * University Of Minnesota - [http://umediblelandscape.blogspot.com/ UM Edible Landscape] * Scripps College - [http://sustainability.claremont.edu/food/index.html Food and Sustainability Community Project] [[Category:Architecture]] [[Category:Permaculture]] [[Category:Universities]] 9e2ia2ig7z2ygyzqiozdy650pk8rihs 0 139384 2819651 2819555 2026-07-28T14:00:22Z Young1lim 21186 /* Adder */ 2819651 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.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260728.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20260728.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] || || [[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]] 38eq1wr24uitsqo2uizkzu1lj7b870k 0 146658 2819698 2625085 2026-07-28T18:37:20Z Lbeaumont 278565 /* General Knowledge Questions */ Removed excessive math examples 2819698 wikitext text/x-wiki ==General Knowledge Questions== Choose a few of these questions to research as a test of your own Theory of Knowledge. Several of these questions are specific to the United States. If you are from another country, substitute analogous questions that pertain to your own country. Some have simple, well accepted answers, some are controversial, and others ask for your opinions. #What is 1+1? #What is 6*3? #What is 2+2*2? #What is 230-220/2? #What is 7+7/7+7*7-7? #What is the missing number in: 16, 6, 68, 88, _, 98? #How old is the universe? #How old is the earth? #When did life begin on earth? #When did the first humans begin life on earth? #Where did the first humans begin life on earth? #Describe, briefly, your understanding of the theory of evolution. Do you believe in that theory? Why or why not? #What, if anything, happens after death? #In what year did the world population first reach one billion people? In what year did it reach six billion people? In what year did it reach seven billion people? #What is the current human population of the earth? How many people, worldwide, are Christians? #Describe your understanding of ''peak oil''. When do you estimate we will reach worldwide peak oil, if ever? #Describe the distinction between petroleum ''reserves'' and petroleum ''resources''. Are world-wide petroleum resources increasing, decreasing, or staying the same? #Describe your understanding of net energy gain. #From what country does the US import most of its oil? #Does the US export any fossil fuels? If so, please describe this. #What role, if any, should ''thorium'' play in our energy policy? #How many people have died, worldwide, as a result of a nuclear power accident? #Many years after an acorn is planted it grows to become a huge oak tree. Where does most of the dry mass of that tree come from? #What establishes the basis for private ownership of land? #Describe your understanding of ''economic externalities''. Give several examples relevant to today’s economy. #Describe your understating of the term “the commons.” Identify several examples of “the commons” that are relevant today. Describe your understating of “the tragedy of the commons.” Describe several examples that are relevant today. What specific proposals, if any, do you have for avoiding actual or potential tragedy of those commons? #Describe the concept of ''Gross Domestic Product'' (GDP) Give examples of destructive activities that are included in GDP. Give examples of pro-social activities that are excluded from the GDP. #Describe your understanding of ''Trickle-down economics''. What evidence supports the theory? What evidence contradicts the theory? #Describe the fallacy of “false dichotomy.” Give several relevant examples. #Describe the difference between ''correlation'' and ''causation''. #Describe the difference between ''anecdotal evidence'' and ''systematic data''. #Describe how a literal truth can send a false message. Give examples. #What, if anything, establishes a fact? How does it differ, if at all, from an opinion? #Describe how you sort through conflicting information to decide what you believe; i.e. describe your ''theory of knowledge''. #What fraction of pregnancies are mistimed, unplanned or unwanted at the time of conception? #Do you believe that contraception prevents abortion? Why or why not? #Are corporations people? Why or why not? #Where does the US rank, worldwide, on various measures of gun violence? #Describe the recent history of gun violence in Australia. #Discuss your understanding of the statement “if guns are outlawed, then only outlaws would have guns.” How is this relevant, if at all, to discussions of gun violence? #What is the current level of gun violence in the United States? Do you find this acceptable? If not, what specific proposals do you have for reducing gun violence? #Are there any limits to the second amendment? If so, please describe them. #Rank the ten amendments in the bill of rights in order of importance. #How many times does the word “Freedom” appear in the US constitution? #Describe the “Great Compromise of 1787” and its role in drafting the US constitution. #Describe the “Three-Fifths Compromise” and its role in drafting the US constitution. #Describe your understanding of ''statesmanship''. Who was the greatest statesman ever? Who is the greatest statesman now serving in congress? #Describe the purpose of federal form I-9. #What role, if any, does form I-9 currently play in immigration policy? What role would you like to see it playing in the future? #How is money created and introduced (i.e. by the central bank) into your nation's monetary system? #What was the federal income tax rate for the highest earners (the top tax bracket) in 1960? What is it now? #What is the purpose of taxation? Is today’s tax system fair? Why or why not? Describe, in detail, a tax system that you believe would be fair. #What is fractional-reserve banking? #Who invented the electric light bulb? #Describe the concept of ''class warfare''. How is it relevant today, if at all? #Describe the distribution of income and wealth in the United States. #Politicians often talk about ''values''. Please rank order the following values from the most important to the least important: 1) Caring for others, 2) Fairness, 3) Liberty, 4) Respect for authority, 5) Loyalty, 6) Sanctity. #Do you believe the current political incumbents propose or enact policies or legislation that violates the US constitution? If so, please cite a specific example of such policies, and identify the section of the constitution that is violated. #Describe your understanding of the phrase “all men are created equal.” Describe how it is relevant to today’s social issues. #Is homosexuality a sinful choice? #Does vaccination improve public health? #Describe the practice of Sharia law in Dearborn, Michigan. In what regions of the United States is Sharia Law practiced or enforced? #In what country was Barack Obama born? #How many people have been deported from the United States during Barack Obama's presidency? How does this compare to other presidencies? #How many people were killed in shark attacks last year? How many died in automobile accidents? #What are the top five causes of death in the United States? Worldwide? #What caused the collapse of the twin towers of the World Trade Center on September 11, 2001? #What is the cost of the F-22 Raptor program? What is the military role of the F-22 Raptor? What is the combat history of the F-22 Raptor? #What fraction of the US federal budget is the defense budget? #How many nuclear weapons exist in each country? #What fraction of the federal budget is devoted to foreign aid and which 3 countries receive the most? # What are the actual goals of the [[w:High Frequency Active Auroral Research Program|High Frequency Active Auroral Research Program]]? #Where does the US education system rank among countries in the developing world? #What countries rank below (have lower death rate) the US in infant mortality rate (calculated as deaths of infants under one year old)? #Where does the US rank in health care outcomes? #Describe ''poverty'' in the United States. #Describe ''malnutrition'' in the United States. #What countries have not adopted SI units (International System of Units) as their official system of weights and measures? #Describe the ''National Bridge inventory'' and the 0-9 structural evaluation scale it uses. How many bridges are rated below 4 on this scale? How many bridges in the United States are classified as “structurally deficient”? How many are classified as “functionally obsolete”? To what extent, if any, does this represent a safety risk? #What are the five most important issues facing the American people? Why do you choose these? What solutions can you offer? # Is faith a reliable epistemology? # What is the historical and archaeological evidence regarding the biblical story of the Exodus? # What is the efficacy of homeopathy? 8zakrtitaxr1ur9xzvceymsny7ycc6m 0 170972 2819730 2581326 2026-07-29T01:37:38Z ~2026-41972-20 3103950 2819730 wikitext text/x-wiki American Sign Language (ASL) is the predominant sign language of deaf communities in the United States and most of anglophone Canada.<ref>[[Wikipedia: American Sign Language]]</ref>. == See Also == * [[Wikipedia: American Sig Language]] == References == {{Reflist}} [[Category:American Sign Language]] 22f9fp41wufs5r6h9x05ywd70a94q6y 2819731 2819730 2026-07-29T01:37:57Z ~2026-41972-20 3103950 2819731 wikitext text/x-wiki American Sign Language (ASL) is the predominant sign language of deaf communities in the United States and most of anglophone Canada.<ref>[[Wikipedia: American Sign Language]]</ref>. == See Also == * [[Wikipedia: American Sign Language]] == References == {{Reflist}} [[Category:American Sign Language]] rlqi1ssj7etkxw3apolob1z27j408is 0 171005 2819656 2819563 2026-07-28T14:16:06Z Young1lim 21186 /* Geometric Series Examples */ 2819656 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.20260728.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]] 04pbqsdmjbreruh021790qbxk50gl9x 0 203942 2819645 2819465 2026-07-28T12:39:36Z Young1lim 21186 /* Lambda Calculus */ 2819645 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260728.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] j3t2b4r02fcszz095t6acf5llv2amtg 0 221601 2819732 2819223 2026-07-29T01:53:40Z Jtneill 10242 /* Instructions */ 2819732 wikitext text/x-wiki {{title|Topic development — Guidelines}} <div style="text-align: center;">''Develop a chapter plan and user page'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 08 |day = 14 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester -->{{:Motivation and emotion/Assessment/Chapter/Contents}}</div> {{TOCright}} ==Overview== * Weight: 10% * Due {{/Due}} * Tasks ** Create a Wikiversity user account. ** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]]. ** Build wiki editing skills. ** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of: *** Title and sub-title *** Headings (and possibly sub-headings) *** Key points for each section (and sub-section) *** Figure (at least 1) *** Learning feature (plan at least 1) *** References (6+ relevant, high quality soruces) *** Resources (2+ see also and 2+ external links) ** Create a Wikiversity user page: *** Introduce yourself *** Summarise at least three different types of social contributions on your Wikiversity user page * Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]]. * Guidance for this assignment is provided in Module 1: ** [[Motivation and emotion/Lectures/Introduction|Lecture 01]] ** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric]. *Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4). * Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page. *Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions. ==Extensions and late submissions== * Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence. * Submissions will be accepted up to 3 days late (-10% per day late). * If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended. ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic. |- style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- ! style="width:20%;" | Category ! style="width:20%;" | Graduate attribute ! style="width:60%;" | Assessment task |- | rowspan="2" style="vertical-align:top;" | '''Be professional''' | style="vertical-align:top;" | Communicate effectively | style="vertical-align:top;" | Communicate your ideas by sharing a chapter plan; provide feedback on other plans. |- | style="vertical-align:top;" | Display initiative and drive, and use organisation skills to plan and manage workload | style="vertical-align:top;" | Get organised by selecting a topic and submitting an on-time chapter plan. |- | style="vertical-align:top;" | '''Be a lifelong learner''' | style="vertical-align:top;" | Evaluate and adopt new technology | style="vertical-align:top;" | Learn how to edit in a collaborative, online environment. |} ==Instructions== Follow these instructions for the topic development: * Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of: *# Title and sub-title (pre-approved or negotiated) *# Overview *# 3–5 other top-level headings *#* Key points for each heading/sub-heading with citations *#* 1+ relevant figure(s) *#* 1+ actual or planned learning feature *# Conclusion *# See also *#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article) *# References (at least 6, which are cited) *# External links *#* 2+ external links (to external resources) *# Wikiversity user page *#* Self-introduction *#* A link to the chapter being worked on *#* Social contributions in a numbered list with a summary and direct link to evidence: *#** 1 direct edit to improve another book chapter (past or present) *#** 1 talk page comment on another book chapter (past or present) *#** 1 {{Motivation and emotion/Canvas}} discussion post * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * <span id="Word count">Length (Word count):</span> There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700). * Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments ==Template== {{:Motivation and emotion/Assessment/Topic/Quickstarttip}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] Topic developments will be marked against the following criteria. {{anchor|Title}} ===Title and sub-title (10%)=== * Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}}) * Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] * Do not include user name; authorship is as per the page's editing history {{anchor|Headings}} ===Headings (10%)=== * Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links) * Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading. * The top-level headings should align with the sub-title and focus questions * Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]]) {{anchor|Overview}} ===Overview (10%)=== * A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] * At least 3 bullet points outlining the "problem" (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]] * 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] {{anchor|Key points}} ===Key points (10%)=== * At least 3 bullet points per section (i.e., per heading or sub-heading) * Overview the most relevant theory(ies), including key citations * Overview the most relevant research, including key citations * Provide at least 1 introductory bullet point before branching into sub-sections * Address the problem (i.e., answer the question in the sub-title) {{Anchor|Figure}} ===Figure (10%)=== * Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]]. * Number each figure sequentially (e.g., Figure 1, Figure 2 etc.) * Include a descriptive caption that connects the figure to the text * Cite each figure at least once in the main text (e.g., see Figure 1) * Optimise image display size to make it easy to read (i.e., not too big or too small) {{Anchor|Learning feature}} ===Learning feature (10%)=== * In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,: ** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] ** Internal (wiki) links: *** At least 1 embedded link to a relevant book chapter *** At least 1 embedded link to a relevant Wikipedia article * Quiz question with correct and incorrect answers ** Table with an APA style caption {{anchor|References}} ===References (10%)=== * Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]]) * Each source should be cited at least once in the key points * Include a balance of key theoretical and key research articles {{anchor|Resources}} ===Resources (10%)=== * '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article) ** Provide at least 1 bullet-pointed: *** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter *** internal wiki link to a relevant Wikipedia page ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., Book chapter, 2023) ** Use alphabetical order * '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources ** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles) ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., The Conversation) ** Use alphabetical order {{anchor|User page}} ===User page (10%)=== * Create a Wikiversity user page for your user account * Edit the user page to provide information about yourself * Recommended headings: ** About me ** Book chapter I'm working on *** Include an internal (wiki) link to the chapter page ** Social contributions * Consider linking to your other online profiles {{anchor|Social contribution}} {{anchor|Socialcontribution}} ===Social contribution (10%)=== * On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions: *# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics) *# provided feedback by commenting on a book chapter's talk page (current or previous topics) *# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum<!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --> * [[Motivation and emotion/Wikiversity/Social contributions|More info]] == Marking rubric== {{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}} ==Examples== ;About * Below are some examples of topic development submissions which received 100% * The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions * It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter) ;2025 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&oldid=2726043 Elina.jean.r] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&oldid=2727975 U3254978] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&oldid=2739659 U3237996] ;2024 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&oldid=2644098 U3216883] ;2023 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&oldid=2550752 U3227684] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&oldid=2552580 U3223114] <!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides. ;2022 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&oldid=2420008 U3203545] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&oldid=2419862 U3210431] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&oldid=2420416 U3216256] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&oldid=2419390 Icantchooseone] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&oldid=2420715 U3217287] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&oldid=2418874 Molzaroid] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&oldid=2420232 Ana028] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&oldid=2418246 U3211603] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&oldid=2417630 U3223109] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&oldid=2420956 U3213441] ;2021 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&action=history U3186377] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&action=history U3202904] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&oldid=2313868 U3194166] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&oldid=2313844 U3187741] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&oldid=2310813 U3187874] * [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&oldid=2312677 U3202854] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&oldid=2311466 U3187178] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&oldid=2311388 J.Payten] ;2020 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&oldid=2191350 U3170940] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&oldid=2198005 U3190210] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&oldid=2198079 U3160224] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&oldid=2194797 U3190467] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&oldid=2199869 NUMBLA0371] ;2019 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&oldid=2051716 U3172958] ;2018 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&oldid=1915671 U3160212] ;2017 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&oldid=1730836 U3122707] --> ==Licensing== Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener. ==See also== * Structure ** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]] ** [[/Checklist|Topic development — Checklist]] * Marking and feedback ** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]] ** [[Template:METF|Feedback template]] ** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric] * Tutorials ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Topic| ]] [[Category:Motivation and emotion guidelines]] qs71t6j60cv5qx3qu2bh1706jkvsfv6 2819734 2819732 2026-07-29T01:58:40Z Jtneill 10242 + links to [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] 2819734 wikitext text/x-wiki {{title|Topic development — Guidelines}} <div style="text-align: center;">''Develop a chapter plan and user page'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 08 |day = 14 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester -->{{:Motivation and emotion/Assessment/Chapter/Contents}}</div> {{TOCright}} ==Overview== * Weight: 10% * Due {{/Due}} * Tasks ** Create a Wikiversity user account. ** Select or [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]]. ** Build wiki editing skills. ** Develop a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] on Wikiversity which consists of: *** Title and sub-title *** Headings (and possibly sub-headings) *** Key points for each section (and sub-section) *** Figure (at least 1) *** Learning feature (plan at least 1) *** References (6+ relevant, high quality soruces) *** Resources (2+ see also and 2+ external links) ** Create a Wikiversity user page: *** Introduce yourself *** Summarise at least three different types of social contributions on your Wikiversity user page * Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]]. * Guidance for this assignment is provided in Module 1: ** [[Motivation and emotion/Lectures/Introduction|Lecture 01]] ** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] and [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing marking rubric]. *Marks will be provided via {{Motivation and emotion/Canvas}} by Census Date (end of Week 4). * Written feedback to help guide [[Motivation and emotion/Assessment/Chapter|book chapter]] drafting will be provided via the topic's Wikiversity discussion page. *Follow up with the [[Motivation and emotion/About/Staff|unit convener]] if you have any questions. ==Extensions and late submissions== * Apply for extension using the unit's online Extension Application Form (see {{Motivation and emotion/Canvas}}) with appropriate documentary evidence. * Submissions will be accepted up to 3 days late (-10% per day late). * If you don't submit this assessment, withdrawal from the unit by Census Date (end of Week 4) is recommended. ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic. |- style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- ! style="width:20%;" | Category ! style="width:20%;" | Graduate attribute ! style="width:60%;" | Assessment task |- | rowspan="2" style="vertical-align:top;" | '''Be professional''' | style="vertical-align:top;" | Communicate effectively | style="vertical-align:top;" | Communicate your ideas by sharing a chapter plan; provide feedback on other plans. |- | style="vertical-align:top;" | Display initiative and drive, and use organisation skills to plan and manage workload | style="vertical-align:top;" | Get organised by selecting a topic and submitting an on-time chapter plan. |- | style="vertical-align:top;" | '''Be a lifelong learner''' | style="vertical-align:top;" | Evaluate and adopt new technology | style="vertical-align:top;" | Learn how to edit in a collaborative, online environment. |} ==Instructions== Follow these instructions for the topic development: * Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of: *# Title and sub-title (pre-approved or [[Motivation and emotion/Assessment/Selection#New topics|negotiated]]) *# Overview *# 3–5 other top-level headings *#* Key points for each heading/sub-heading with citations *#* 1+ relevant figure(s) *#* 1+ actual or planned learning feature *# Conclusion *# See also *#* 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article) *# References (at least 6, which are cited) *# External links *#* 2+ external links (to external resources) *# Wikiversity user page *#* Self-introduction *#* A link to the chapter being worked on *#* Social contributions in a numbered list with a summary and direct link to evidence: *#** 1 direct edit to improve another book chapter (past or present) *#** 1 talk page comment on another book chapter (past or present) *#** 1 {{Motivation and emotion/Canvas}} discussion post * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * <span id="Word count">Length (Word count):</span> There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700). * Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments ==Template== {{:Motivation and emotion/Assessment/Topic/Quickstarttip}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] Topic developments will be marked against the following criteria. {{anchor|Title}} ===Title and sub-title (10%)=== * Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}}) * Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] * Do not include user name; authorship is as per the page's editing history {{anchor|Headings}} ===Headings (10%)=== * Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links) * Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading. * The top-level headings should align with the sub-title and focus questions * Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]]) {{anchor|Overview}} ===Overview (10%)=== * A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] * At least 3 bullet points outlining the "problem" (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]] * 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] {{anchor|Key points}} ===Key points (10%)=== * At least 3 bullet points per section (i.e., per heading or sub-heading) * Overview the most relevant theory(ies), including key citations * Overview the most relevant research, including key citations * Provide at least 1 introductory bullet point before branching into sub-sections * Address the problem (i.e., answer the question in the sub-title) {{Anchor|Figure}} ===Figure (10%)=== * Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]]. * Number each figure sequentially (e.g., Figure 1, Figure 2 etc.) * Include a descriptive caption that connects the figure to the text * Cite each figure at least once in the main text (e.g., see Figure 1) * Optimise image display size to make it easy to read (i.e., not too big or too small) {{Anchor|Learning feature}} ===Learning feature (10%)=== * In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,: ** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] ** Internal (wiki) links: *** At least 1 embedded link to a relevant book chapter *** At least 1 embedded link to a relevant Wikipedia article * Quiz question with correct and incorrect answers ** Table with an APA style caption {{anchor|References}} ===References (10%)=== * Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]]) * Each source should be cited at least once in the key points * Include a balance of key theoretical and key research articles {{anchor|Resources}} ===Resources (10%)=== * '''See also''' (Level 2 heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article) ** Provide at least 1 bullet-pointed: *** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter *** internal wiki link to a relevant Wikipedia page ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., Book chapter, 2023) ** Use alphabetical order * '''External links''' (Level 2 heading): Provide at least 2 external links to key internet resources ** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles) ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., The Conversation) ** Use alphabetical order {{anchor|User page}} ===User page (10%)=== * Create a Wikiversity user page for your user account * Edit the user page to provide information about yourself * Recommended headings: ** About me ** Book chapter I'm working on *** Include an internal (wiki) link to the chapter page ** Social contributions * Consider linking to your other online profiles {{anchor|Social contribution}} {{anchor|Socialcontribution}} ===Social contribution (10%)=== * On your Wikiversity user page, summarise and link to ''direct evidence'' that you made at least 3 different types of contributions: *# direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics) *# provided feedback by commenting on a book chapter's talk page (current or previous topics) *# posted about motivation or emotion or the assessment tasks to the {{Motivation and emotion/Canvas}} discussion forum<!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --> * [[Motivation and emotion/Wikiversity/Social contributions|More info]] == Marking rubric== {{Notice|1=[https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric]}} ==Examples== ;About * Below are some examples of topic development submissions which received 100% * The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions * It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter) ;2025 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&oldid=2726043 Elina.jean.r] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&oldid=2727975 U3254978] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&oldid=2739659 U3237996] ;2024 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&oldid=2644098 U3216883] ;2023 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&oldid=2550752 U3227684] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&oldid=2552580 U3223114] <!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides. ;2022 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&oldid=2420008 U3203545] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&oldid=2419862 U3210431] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&oldid=2420416 U3216256] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&oldid=2419390 Icantchooseone] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&oldid=2420715 U3217287] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&oldid=2418874 Molzaroid] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&oldid=2420232 Ana028] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&oldid=2418246 U3211603] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&oldid=2417630 U3223109] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&oldid=2420956 U3213441] ;2021 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&action=history U3186377] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&action=history U3202904] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&oldid=2313868 U3194166] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&oldid=2313844 U3187741] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&oldid=2310813 U3187874] * [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&oldid=2312677 U3202854] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&oldid=2311466 U3187178] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&oldid=2311388 J.Payten] ;2020 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&oldid=2191350 U3170940] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&oldid=2198005 U3190210] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&oldid=2198079 U3160224] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&oldid=2194797 U3190467] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&oldid=2199869 NUMBLA0371] ;2019 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&oldid=2051716 U3172958] ;2018 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&oldid=1915671 U3160212] ;2017 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&oldid=1730836 U3122707] --> ==Licensing== Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener. ==See also== * Structure ** [[Template:Motivation and emotion/Book chapter structure|Book chapter structure template]] ** [[/Checklist|Topic development — Checklist]] * Marking and feedback ** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]] ** [[Template:METF|Feedback template]] ** [https://docs.google.com/document/d/1yQd1-IekznJNLwhef-yEBaxKtRR6dAVESWXoTa-ti3c/edit?usp=sharing Marking rubric] * Tutorials ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Topic| ]] [[Category:Motivation and emotion guidelines]] dkis5liqqux0erbtvop7gwgezf8ats3 0 264907 2819736 2815389 2026-07-29T02:02:41Z Jtneill 10242 + links to negotiate topics 2819736 wikitext text/x-wiki {{Motivation and emotion/Tutorials|Tutorial 1: Topic selection|first}} {{Motivation and emotion/Tutorials/In development}} <!-- {{Motivation and emotion/Tutorials/Complete2}} --> <!-- {{Motivation and emotion/Tutorials/In development}} --> <!-- {{Note|For more info, see [[Motivation and emotion/Tutorials/Introduction]]}} --> ==Overview== This tutorial: # explains how [[Motivation and emotion/Tutorials|tutorials]] will work and what will be covered # shows how to select a topic for the [[Motivation and emotion/Assessment/Major project|major project]] # involves students in sharing about their selected topic or interests # demonstrates how to add a template to help scaffold the topic development ==Welcome== # Welcome and tutor introduction # Tutorial content will typically involve: ## Discussion of key concepts from the textbook and lecture ## Group activities that expand and explore key concepts ## Project work which develops skills for the topic development, book chapter, and exam # Overview of [[Motivation and emotion/Tutorials|tutorial topics]] ;Virtual room tour: # Chat # Microphone (headset recommended) / webcam # Raise hand # Status and feedback # Whiteboard tools # Breakout rooms # Polls ==Icebreaker== These get-to-know-you activities provide an opportunity to meet and greet and start to explore some examples of motivations and emotions. ;Virtual # Speedfriending: 3 minutes to meet, greet, and share about what you're looking forward to (or not) in [[motivation and emotion]] ;In-person # [[w:sociometry|sociometric]] exercise or whiteboard discussion: ## What job/career do you hope to be doing in 5 to 10 years' time? (examples of motivations) ## How have you been feeling so far today? (Mood/Emotion) (examples of emotions) <!-- Ask participants to arrange themselves in a line or groups which represent our similarities and differences in motivation and emotion. For each question: * Ask participants to introduce themselves to the people next to them / in their group. * Link the sociometric data and responses to concepts in motivation and emotion. --> ==Topic check-in and whiparound== ===Check-in=== Poll - What stage are you up to? # Huh? What do I do? # I created a Wikiversity user account # I signed up to a book topic # I have a topic and started editing # I'm all over it! ===Whiparound=== #[https://goalbookapp.com/toolkit/v/strategy/whiparound Whiparound] discussion based on each person's book chapter interests and progress. Ask each student to share: ## their name ## topic they've chosen (or what areas they are interested in) # According to each person's stage of development: ## Very briefly discuss or expand on each topic, suggest related resources or ideas etc. ## If students are unsure about their interests, probe about what they are curious about - or at least find out, are they more interested in motivation or emotion etc.? # Key points: ## Topics are typically allocated to either motivation or emotion (with a few listed under motivation and emotion). Be aware of which broad category the selected topic fits into. ## Some topics are broader (requiring a zoomed out overview without getting too deep in the weeds) and others are narrower (requiring more comprehensive, indepth coverage). ## Broader topics can be overwhelming if there is a lot of material; an option in this case is to [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] a narrowing of the topic title and/or sub-title. ## Narrower topics for which there is a lack of sufficient literature may benefit from negotiating a broadening of the topic title and/or sub-title. ==Project tasks== ===Create a Wikiversity account=== # [[Create an account|Create an account]]: ## Register a user name and password. Include an email for password recovery. ## Edits to Wikiversity are publicly available, so choose a name that suits your privacy requirements (e.g., can be a real name, pseudonym, or student number). ## Start at http://en.wikiversity.org (English Wikiversity) - Create account (top-right) ## Or go direct to: [[Special:Userlogin/signup|create a Wikiversity user account]] # Log in (top-right) ===Topic selection=== # Topic selection is [[wikt:first in, best dressed|first in, best dressed]] # Sign up to a pre-approved topic in the {{Motivation and emotion/Book}} by: ## Clicking "Edit" or "Edit source" ## Replace "User name" alongside your topic of choice with your Wikiversity user name ## Publish the page ## Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a topic]] ===Quick start template=== # It is recommended (but not required) that you add the [[Template:Motivation and emotion/Book chapter structure|quick start template]] to the topic page. # To do this, click "Create source" or "Edit source" on the target page, paste '''[[Template:Motivation and emotion/Book chapter structure|<nowiki>{{subst:ME/BCS}}</nowiki>]]''', then Publish. ## If you get a spam filter warning it is probably because you have a new account. You can ask the tutor to add the template for you. Or publish some smaller edits first (e.g., on your user page). This helps the system learn to trust you, then try inserting the template again. ==Next steps== # Sign up for (or [[Motivation and emotion/Assessment/Selection#New topics|negotiate]]) a book chapter topic by the end of Week 2. # Draft the [[Motivation and emotion/Assessment/Topic|topic development]] which is a detailed plan for a full chapter, consisting of main headings with bullet-points about key points, on Wikiversity by the end of Week 3. ==Recording== <!-- * [ Tutorial 1] (2026) --> * [https://au-lti.bbcollab.com/recording/71cf60b37faf4d40b17f9107a417e0ee Tutorial 1] (2025)<!-- * [https://au-lti.bbcollab.com/recording/81cb0a731ef44fe1ab82b529fd3a836a Tutorial 1] (2024) * [https://au-lti.bbcollab.com/recording/ace491a49cdd4d8f8441c7b357fa60dc Tutorial 1] (2023) * [https://au-lti.bbcollab.com/recording/91c93320558847a8a6717e2a0ad4b6dd Tutorial 1 recording] (2022) * [https://au-lti.bbcollab.com/recording/332ed2ad09824fb88c799b92cd64eaaa Tutorial 1 recording] (2021) --> ==See also== ;Topic development * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] ;List of topics # The {{Motivation and emotion/Book}} ;Lecture * [[Motivation and emotion/Lectures/Introduction|Introduction]] ;Tutorial * [[{{#titleparts:{{PAGENAME}}|2}}/Wiki editing|Wiki editing]] (Next tutorial) <!-- ;Admin * [[/Instructor notes/]] * [[Introduction to psychology/Psy102/Tutorials/Motivation|Motivation tutorial]] (Psychology 102) --> <noinclude>{{Motivation and emotion/Tutorials/Navigation}}</noinclude> [[Category:Motivation and emotion/Tutorials/Topic selection]] h6m5wbacyw37scm5xu4kz27s8bjtu1m 2819738 2819736 2026-07-29T02:05:09Z Jtneill 10242 External links 2819738 wikitext text/x-wiki {{Motivation and emotion/Tutorials|Tutorial 1: Topic selection|first}} {{Motivation and emotion/Tutorials/In development}} <!-- {{Motivation and emotion/Tutorials/Complete2}} --> <!-- {{Motivation and emotion/Tutorials/In development}} --> <!-- {{Note|For more info, see [[Motivation and emotion/Tutorials/Introduction]]}} --> ==Overview== This tutorial: # explains how [[Motivation and emotion/Tutorials|tutorials]] will work and what will be covered # shows how to select a topic for the [[Motivation and emotion/Assessment/Major project|major project]] # involves students in sharing about their selected topic or interests # demonstrates how to add a template to help scaffold the topic development ==Welcome== # Welcome and tutor introduction # Tutorial content will typically involve: ## Discussion of key concepts from the textbook and lecture ## Group activities that expand and explore key concepts ## Project work which develops skills for the topic development, book chapter, and exam # Overview of [[Motivation and emotion/Tutorials|tutorial topics]] ;Virtual room tour: # Chat # Microphone (headset recommended) / webcam # Raise hand # Status and feedback # Whiteboard tools # Breakout rooms # Polls ==Icebreaker== These get-to-know-you activities provide an opportunity to meet and greet and start to explore some examples of motivations and emotions. ;Virtual # Speedfriending: 3 minutes to meet, greet, and share about what you're looking forward to (or not) in [[motivation and emotion]] ;In-person # [[w:sociometry|sociometric]] exercise or whiteboard discussion: ## What job/career do you hope to be doing in 5 to 10 years' time? (examples of motivations) ## How have you been feeling so far today? (Mood/Emotion) (examples of emotions) <!-- Ask participants to arrange themselves in a line or groups which represent our similarities and differences in motivation and emotion. For each question: * Ask participants to introduce themselves to the people next to them / in their group. * Link the sociometric data and responses to concepts in motivation and emotion. --> ==Topic check-in and whiparound== ===Check-in=== Poll - What stage are you up to? # Huh? What do I do? # I created a Wikiversity user account # I signed up to a book topic # I have a topic and started editing # I'm all over it! ===Whiparound=== #[https://goalbookapp.com/toolkit/v/strategy/whiparound Whiparound] discussion based on each person's book chapter interests and progress. Ask each student to share: ## their name ## topic they've chosen (or what areas they are interested in) # According to each person's stage of development: ## Very briefly discuss or expand on each topic, suggest related resources or ideas etc. ## If students are unsure about their interests, probe about what they are curious about - or at least find out, are they more interested in motivation or emotion etc.? # Key points: ## Topics are typically allocated to either motivation or emotion (with a few listed under motivation and emotion). Be aware of which broad category the selected topic fits into. ## Some topics are broader (requiring a zoomed out overview without getting too deep in the weeds) and others are narrower (requiring more comprehensive, indepth coverage). ## Broader topics can be overwhelming if there is a lot of material; an option in this case is to [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] a narrowing of the topic title and/or sub-title. ## Narrower topics for which there is a lack of sufficient literature may benefit from negotiating a broadening of the topic title and/or sub-title. ==Project tasks== ===Create a Wikiversity account=== # [[Create an account|Create an account]]: ## Register a user name and password. Include an email for password recovery. ## Edits to Wikiversity are publicly available, so choose a name that suits your privacy requirements (e.g., can be a real name, pseudonym, or student number). ## Start at http://en.wikiversity.org (English Wikiversity) - Create account (top-right) ## Or go direct to: [[Special:Userlogin/signup|create a Wikiversity user account]] # Log in (top-right) ===Topic selection=== # Topic selection is [[wikt:first in, best dressed|first in, best dressed]] # Sign up to a pre-approved topic in the {{Motivation and emotion/Book}} by: ## Clicking "Edit" or "Edit source" ## Replace "User name" alongside your topic of choice with your Wikiversity user name ## Publish the page ## Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a topic]] ===Quick start template=== # It is recommended (but not required) that you add the [[Template:Motivation and emotion/Book chapter structure|quick start template]] to the topic page. # To do this, click "Create source" or "Edit source" on the target page, paste '''[[Template:Motivation and emotion/Book chapter structure|<nowiki>{{subst:ME/BCS}}</nowiki>]]''', then Publish. ## If you get a spam filter warning it is probably because you have a new account. You can ask the tutor to add the template for you. Or publish some smaller edits first (e.g., on your user page). This helps the system learn to trust you, then try inserting the template again. ==Next steps== # Sign up for (or [[Motivation and emotion/Assessment/Selection#New topics|negotiate]]) a book chapter topic by the end of Week 2. # Draft the [[Motivation and emotion/Assessment/Topic|topic development]] which is a detailed plan for a full chapter, consisting of main headings with bullet-points about key points, on Wikiversity by the end of Week 3. ==Recording== <!-- * [ Tutorial 1] (2026) --> * [https://au-lti.bbcollab.com/recording/71cf60b37faf4d40b17f9107a417e0ee Tutorial 1] (2025)<!-- * [https://au-lti.bbcollab.com/recording/81cb0a731ef44fe1ab82b529fd3a836a Tutorial 1] (2024) * [https://au-lti.bbcollab.com/recording/ace491a49cdd4d8f8441c7b357fa60dc Tutorial 1] (2023) * [https://au-lti.bbcollab.com/recording/91c93320558847a8a6717e2a0ad4b6dd Tutorial 1 recording] (2022) * [https://au-lti.bbcollab.com/recording/332ed2ad09824fb88c799b92cd64eaaa Tutorial 1 recording] (2021) --> ==See also== ;Topic development * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] ;List of topics # The {{Motivation and emotion/Book}} ;Lecture * [[Motivation and emotion/Lectures/Introduction|Introduction]] ;Tutorial * [[{{#titleparts:{{PAGENAME}}|2}}/Wiki editing|Wiki editing]] (Next tutorial) <!-- ;Admin * [[/Instructor notes/]] * [[Introduction to psychology/Psy102/Tutorials/Motivation|Motivation tutorial]] (Psychology 102) --> ==External links== * [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti) <noinclude>{{Motivation and emotion/Tutorials/Navigation}}</noinclude> [[Category:Motivation and emotion/Tutorials/Topic selection]] enh6c51nz40zzd25g1xa18et3afw79g 2819739 2819738 2026-07-29T02:06:02Z Jtneill 10242 /* See also */ 2819739 wikitext text/x-wiki {{Motivation and emotion/Tutorials|Tutorial 1: Topic selection|first}} {{Motivation and emotion/Tutorials/In development}} <!-- {{Motivation and emotion/Tutorials/Complete2}} --> <!-- {{Motivation and emotion/Tutorials/In development}} --> <!-- {{Note|For more info, see [[Motivation and emotion/Tutorials/Introduction]]}} --> ==Overview== This tutorial: # explains how [[Motivation and emotion/Tutorials|tutorials]] will work and what will be covered # shows how to select a topic for the [[Motivation and emotion/Assessment/Major project|major project]] # involves students in sharing about their selected topic or interests # demonstrates how to add a template to help scaffold the topic development ==Welcome== # Welcome and tutor introduction # Tutorial content will typically involve: ## Discussion of key concepts from the textbook and lecture ## Group activities that expand and explore key concepts ## Project work which develops skills for the topic development, book chapter, and exam # Overview of [[Motivation and emotion/Tutorials|tutorial topics]] ;Virtual room tour: # Chat # Microphone (headset recommended) / webcam # Raise hand # Status and feedback # Whiteboard tools # Breakout rooms # Polls ==Icebreaker== These get-to-know-you activities provide an opportunity to meet and greet and start to explore some examples of motivations and emotions. ;Virtual # Speedfriending: 3 minutes to meet, greet, and share about what you're looking forward to (or not) in [[motivation and emotion]] ;In-person # [[w:sociometry|sociometric]] exercise or whiteboard discussion: ## What job/career do you hope to be doing in 5 to 10 years' time? (examples of motivations) ## How have you been feeling so far today? (Mood/Emotion) (examples of emotions) <!-- Ask participants to arrange themselves in a line or groups which represent our similarities and differences in motivation and emotion. For each question: * Ask participants to introduce themselves to the people next to them / in their group. * Link the sociometric data and responses to concepts in motivation and emotion. --> ==Topic check-in and whiparound== ===Check-in=== Poll - What stage are you up to? # Huh? What do I do? # I created a Wikiversity user account # I signed up to a book topic # I have a topic and started editing # I'm all over it! ===Whiparound=== #[https://goalbookapp.com/toolkit/v/strategy/whiparound Whiparound] discussion based on each person's book chapter interests and progress. Ask each student to share: ## their name ## topic they've chosen (or what areas they are interested in) # According to each person's stage of development: ## Very briefly discuss or expand on each topic, suggest related resources or ideas etc. ## If students are unsure about their interests, probe about what they are curious about - or at least find out, are they more interested in motivation or emotion etc.? # Key points: ## Topics are typically allocated to either motivation or emotion (with a few listed under motivation and emotion). Be aware of which broad category the selected topic fits into. ## Some topics are broader (requiring a zoomed out overview without getting too deep in the weeds) and others are narrower (requiring more comprehensive, indepth coverage). ## Broader topics can be overwhelming if there is a lot of material; an option in this case is to [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] a narrowing of the topic title and/or sub-title. ## Narrower topics for which there is a lack of sufficient literature may benefit from negotiating a broadening of the topic title and/or sub-title. ==Project tasks== ===Create a Wikiversity account=== # [[Create an account|Create an account]]: ## Register a user name and password. Include an email for password recovery. ## Edits to Wikiversity are publicly available, so choose a name that suits your privacy requirements (e.g., can be a real name, pseudonym, or student number). ## Start at http://en.wikiversity.org (English Wikiversity) - Create account (top-right) ## Or go direct to: [[Special:Userlogin/signup|create a Wikiversity user account]] # Log in (top-right) ===Topic selection=== # Topic selection is [[wikt:first in, best dressed|first in, best dressed]] # Sign up to a pre-approved topic in the {{Motivation and emotion/Book}} by: ## Clicking "Edit" or "Edit source" ## Replace "User name" alongside your topic of choice with your Wikiversity user name ## Publish the page ## Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a topic]] ===Quick start template=== # It is recommended (but not required) that you add the [[Template:Motivation and emotion/Book chapter structure|quick start template]] to the topic page. # To do this, click "Create source" or "Edit source" on the target page, paste '''[[Template:Motivation and emotion/Book chapter structure|<nowiki>{{subst:ME/BCS}}</nowiki>]]''', then Publish. ## If you get a spam filter warning it is probably because you have a new account. You can ask the tutor to add the template for you. Or publish some smaller edits first (e.g., on your user page). This helps the system learn to trust you, then try inserting the template again. ==Next steps== # Sign up for (or [[Motivation and emotion/Assessment/Selection#New topics|negotiate]]) a book chapter topic by the end of Week 2. # Draft the [[Motivation and emotion/Assessment/Topic|topic development]] which is a detailed plan for a full chapter, consisting of main headings with bullet-points about key points, on Wikiversity by the end of Week 3. ==Recording== <!-- * [ Tutorial 1] (2026) --> * [https://au-lti.bbcollab.com/recording/71cf60b37faf4d40b17f9107a417e0ee Tutorial 1] (2025)<!-- * [https://au-lti.bbcollab.com/recording/81cb0a731ef44fe1ab82b529fd3a836a Tutorial 1] (2024) * [https://au-lti.bbcollab.com/recording/ace491a49cdd4d8f8441c7b357fa60dc Tutorial 1] (2023) * [https://au-lti.bbcollab.com/recording/91c93320558847a8a6717e2a0ad4b6dd Tutorial 1 recording] (2022) * [https://au-lti.bbcollab.com/recording/332ed2ad09824fb88c799b92cd64eaaa Tutorial 1 recording] (2021) --> ==See also== ;Topic development * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] ;List of topics # {{Motivation and emotion/Book}} ;Lecture * [[Motivation and emotion/Lectures/Introduction|Introduction]] ;Tutorial * [[{{#titleparts:{{PAGENAME}}|2}}/Wiki editing|Wiki editing]] (Next tutorial) <!-- ;Admin * [[/Instructor notes/]] * [[Introduction to psychology/Psy102/Tutorials/Motivation|Motivation tutorial]] (Psychology 102) --> ==External links== * [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti) <noinclude>{{Motivation and emotion/Tutorials/Navigation}}</noinclude> [[Category:Motivation and emotion/Tutorials/Topic selection]] 6q07xzdody86br1zn7ffzw3enmqfwm1 2819740 2819739 2026-07-29T02:07:14Z Jtneill 10242 /* See also */ 2819740 wikitext text/x-wiki {{Motivation and emotion/Tutorials|Tutorial 1: Topic selection|first}} {{Motivation and emotion/Tutorials/In development}} <!-- {{Motivation and emotion/Tutorials/Complete2}} --> <!-- {{Motivation and emotion/Tutorials/In development}} --> <!-- {{Note|For more info, see [[Motivation and emotion/Tutorials/Introduction]]}} --> ==Overview== This tutorial: # explains how [[Motivation and emotion/Tutorials|tutorials]] will work and what will be covered # shows how to select a topic for the [[Motivation and emotion/Assessment/Major project|major project]] # involves students in sharing about their selected topic or interests # demonstrates how to add a template to help scaffold the topic development ==Welcome== # Welcome and tutor introduction # Tutorial content will typically involve: ## Discussion of key concepts from the textbook and lecture ## Group activities that expand and explore key concepts ## Project work which develops skills for the topic development, book chapter, and exam # Overview of [[Motivation and emotion/Tutorials|tutorial topics]] ;Virtual room tour: # Chat # Microphone (headset recommended) / webcam # Raise hand # Status and feedback # Whiteboard tools # Breakout rooms # Polls ==Icebreaker== These get-to-know-you activities provide an opportunity to meet and greet and start to explore some examples of motivations and emotions. ;Virtual # Speedfriending: 3 minutes to meet, greet, and share about what you're looking forward to (or not) in [[motivation and emotion]] ;In-person # [[w:sociometry|sociometric]] exercise or whiteboard discussion: ## What job/career do you hope to be doing in 5 to 10 years' time? (examples of motivations) ## How have you been feeling so far today? (Mood/Emotion) (examples of emotions) <!-- Ask participants to arrange themselves in a line or groups which represent our similarities and differences in motivation and emotion. For each question: * Ask participants to introduce themselves to the people next to them / in their group. * Link the sociometric data and responses to concepts in motivation and emotion. --> ==Topic check-in and whiparound== ===Check-in=== Poll - What stage are you up to? # Huh? What do I do? # I created a Wikiversity user account # I signed up to a book topic # I have a topic and started editing # I'm all over it! ===Whiparound=== #[https://goalbookapp.com/toolkit/v/strategy/whiparound Whiparound] discussion based on each person's book chapter interests and progress. Ask each student to share: ## their name ## topic they've chosen (or what areas they are interested in) # According to each person's stage of development: ## Very briefly discuss or expand on each topic, suggest related resources or ideas etc. ## If students are unsure about their interests, probe about what they are curious about - or at least find out, are they more interested in motivation or emotion etc.? # Key points: ## Topics are typically allocated to either motivation or emotion (with a few listed under motivation and emotion). Be aware of which broad category the selected topic fits into. ## Some topics are broader (requiring a zoomed out overview without getting too deep in the weeds) and others are narrower (requiring more comprehensive, indepth coverage). ## Broader topics can be overwhelming if there is a lot of material; an option in this case is to [[Motivation and emotion/Assessment/Selection#New topics|negotiate]] a narrowing of the topic title and/or sub-title. ## Narrower topics for which there is a lack of sufficient literature may benefit from negotiating a broadening of the topic title and/or sub-title. ==Project tasks== ===Create a Wikiversity account=== # [[Create an account|Create an account]]: ## Register a user name and password. Include an email for password recovery. ## Edits to Wikiversity are publicly available, so choose a name that suits your privacy requirements (e.g., can be a real name, pseudonym, or student number). ## Start at http://en.wikiversity.org (English Wikiversity) - Create account (top-right) ## Or go direct to: [[Special:Userlogin/signup|create a Wikiversity user account]] # Log in (top-right) ===Topic selection=== # Topic selection is [[wikt:first in, best dressed|first in, best dressed]] # Sign up to a pre-approved topic in the {{Motivation and emotion/Book}} by: ## Clicking "Edit" or "Edit source" ## Replace "User name" alongside your topic of choice with your Wikiversity user name ## Publish the page ## Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a topic]] ===Quick start template=== # It is recommended (but not required) that you add the [[Template:Motivation and emotion/Book chapter structure|quick start template]] to the topic page. # To do this, click "Create source" or "Edit source" on the target page, paste '''[[Template:Motivation and emotion/Book chapter structure|<nowiki>{{subst:ME/BCS}}</nowiki>]]''', then Publish. ## If you get a spam filter warning it is probably because you have a new account. You can ask the tutor to add the template for you. Or publish some smaller edits first (e.g., on your user page). This helps the system learn to trust you, then try inserting the template again. ==Next steps== # Sign up for (or [[Motivation and emotion/Assessment/Selection#New topics|negotiate]]) a book chapter topic by the end of Week 2. # Draft the [[Motivation and emotion/Assessment/Topic|topic development]] which is a detailed plan for a full chapter, consisting of main headings with bullet-points about key points, on Wikiversity by the end of Week 3. ==Recording== <!-- * [ Tutorial 1] (2026) --> * [https://au-lti.bbcollab.com/recording/71cf60b37faf4d40b17f9107a417e0ee Tutorial 1] (2025)<!-- * [https://au-lti.bbcollab.com/recording/81cb0a731ef44fe1ab82b529fd3a836a Tutorial 1] (2024) * [https://au-lti.bbcollab.com/recording/ace491a49cdd4d8f8441c7b357fa60dc Tutorial 1] (2023) * [https://au-lti.bbcollab.com/recording/91c93320558847a8a6717e2a0ad4b6dd Tutorial 1 recording] (2022) * [https://au-lti.bbcollab.com/recording/332ed2ad09824fb88c799b92cd64eaaa Tutorial 1 recording] (2021) --> ==See also== ;Topic development * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Motivation and emotion/Book/2026|Book chapter table of contents]] (2026) ;Lecture * [[Motivation and emotion/Lectures/Introduction|Introduction]] ;Tutorial * [[{{#titleparts:{{PAGENAME}}|2}}/Wiki editing|Wiki editing]] (Next tutorial) <!-- ;Admin * [[/Instructor notes/]] * [[Introduction to psychology/Psy102/Tutorials/Motivation|Motivation tutorial]] (Psychology 102) --> ==External links== * [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti) <noinclude>{{Motivation and emotion/Tutorials/Navigation}}</noinclude> [[Category:Motivation and emotion/Tutorials/Topic selection]] 5tirzt8uhdi67xmeobup15ad3z50gef 0 285380 2819654 2819558 2026-07-28T14:11:24Z Young1lim 21186 /* Applications */ 2819654 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.20260728.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> ofgnaqfbxvs36dadr87ab6nx8tp2gbs 0 285854 2819733 2726003 2026-07-29T01:56:46Z Jtneill 10242 /* New topics */ [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter generator] (Cogniti) 2819733 wikitext text/x-wiki {{title|Topic selection — Guidelines}} <!--<div style="text-align: center;">''Select major project topic'' --> <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---> <!-- {{countdown |year = 2024 |month = 08 |day = 04 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}</div> --> {{TOCright}} ==Overview== <!-- * Weight: 0% * Due: {{/Due}} * Ungraded early assessment exercise * Tasks --> * [[Special:CreateAccount|Create a Wikiversity account]] * Sign up to (or negotiate) an approved topic for the major project — see {{Motivation and emotion/Book}} * Ask any clarifying questions <!-- ==Marking and feedback== * No marks * Feedback will be provided to: ** Approve topic selection ** Respond to student questions ** Provide suggestions * Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date * Follow up if you don't understand the feedback ==Extensions and late submissions== * No extensions or late submissions * If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]] ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Select an appropriate, unique, specific motivation or emotion topic for the major project |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Be professional — communicate effectively | style="vertical-align:top;" | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions |- | style="vertical-align:top;" | Be professional — display initiative and drive, and use organisation skills to plan and manage workload | style="vertical-align:top;" | Get organised by selecting a topic |- | style="vertical-align:top;" | Be a lifelong learner — evaluate and adopt new technology | style="vertical-align:top;" | Create a Wikiversity account and make at least 1 edit by signing up to a topic |} --> ==Instructions== Create a Wikiversity account and sign up to a pre-approved major project topic in the {{Motivation and emotion/Book}} or negotiate a new topic. This topic will be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]. Ask any clarifying questions. ===[[Special:CreateAccount|Create a Wikiversity account]]=== * User names start with a capital letter * Can use real name, pseudonym, student number etc. ===Sign up to a topic=== '''Pre-approved topics''' are available via the {{Motivation and emotion/Book}} * How to sign up: ** Click "Edit" or "Edit source" ** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name ** Click "Publish" <!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --> * Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]] * You can change topics by moving your username to a topic without an author <!-- * More topics will be added, but students are also encouraged to propose topics --> ===New topics=== '''New topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]]. * Ways to get some ideas: ** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter generator] (Cogniti) ** Look through chapters written in [[Motivation and emotion/Book|previous years]] ** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet * New topics must: ** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work? ** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science ** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]] * To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details: ** Title ** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]] ** Wikiversity user name ** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]]) ===Ask clarifying questions=== * What questions could you ask to help you to successfully tackle the major project? ==See also== * [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Selection| ]] ejky5cnhvo5ahibwtkg1ysj2ygsu67b 2819735 2819733 2026-07-29T01:59:12Z Jtneill 10242 /* New topics */ 2819735 wikitext text/x-wiki {{title|Topic selection — Guidelines}} <!--<div style="text-align: center;">''Select major project topic'' --> <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---> <!-- {{countdown |year = 2024 |month = 08 |day = 04 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}</div> --> {{TOCright}} ==Overview== <!-- * Weight: 0% * Due: {{/Due}} * Ungraded early assessment exercise * Tasks --> * [[Special:CreateAccount|Create a Wikiversity account]] * Sign up to (or negotiate) an approved topic for the major project — see {{Motivation and emotion/Book}} * Ask any clarifying questions <!-- ==Marking and feedback== * No marks * Feedback will be provided to: ** Approve topic selection ** Respond to student questions ** Provide suggestions * Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date * Follow up if you don't understand the feedback ==Extensions and late submissions== * No extensions or late submissions * If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]] ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Select an appropriate, unique, specific motivation or emotion topic for the major project |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Be professional — communicate effectively | style="vertical-align:top;" | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions |- | style="vertical-align:top;" | Be professional — display initiative and drive, and use organisation skills to plan and manage workload | style="vertical-align:top;" | Get organised by selecting a topic |- | style="vertical-align:top;" | Be a lifelong learner — evaluate and adopt new technology | style="vertical-align:top;" | Create a Wikiversity account and make at least 1 edit by signing up to a topic |} --> ==Instructions== Create a Wikiversity account and sign up to a pre-approved major project topic in the {{Motivation and emotion/Book}} or negotiate a new topic. This topic will be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]. Ask any clarifying questions. ===[[Special:CreateAccount|Create a Wikiversity account]]=== * User names start with a capital letter * Can use real name, pseudonym, student number etc. ===Sign up to a topic=== '''Pre-approved topics''' are available via the {{Motivation and emotion/Book}} * How to sign up: ** Click "Edit" or "Edit source" ** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name ** Click "Publish" <!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --> * Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]] * You can change topics by moving your username to a topic without an author <!-- * More topics will be added, but students are also encouraged to propose topics --> ===New topics=== '''New topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]]. * Ways to get some ideas: ** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti) ** Look through chapters written in [[Motivation and emotion/Book|previous years]] ** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet * New topics must: ** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work? ** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science ** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]] * To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details: ** Title ** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]] ** Wikiversity user name ** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]]) ===Ask clarifying questions=== * What questions could you ask to help you to successfully tackle the major project? ==See also== * [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Selection| ]] 53mtwuai5tg0g18ob2my9j2nkxi3fg1 2819737 2819735 2026-07-29T02:03:23Z Jtneill 10242 /* New topics */ 2819737 wikitext text/x-wiki {{title|Topic selection — Guidelines}} <!--<div style="text-align: center;">''Select major project topic'' --> <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---> <!-- {{countdown |year = 2024 |month = 08 |day = 04 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester {{:Motivation and emotion/Assessment/Chapter/Contents}}</div> --> {{TOCright}} ==Overview== <!-- * Weight: 0% * Due: {{/Due}} * Ungraded early assessment exercise * Tasks --> * [[Special:CreateAccount|Create a Wikiversity account]] * Sign up to (or negotiate) an approved topic for the major project — see {{Motivation and emotion/Book}} * Ask any clarifying questions <!-- ==Marking and feedback== * No marks * Feedback will be provided to: ** Approve topic selection ** Respond to student questions ** Provide suggestions * Feedback will be available via {{Motivation and emotion/Canvas}} before the [[Motivation and emotion/Assessment/Topic|topic development]] due date * Follow up if you don't understand the feedback ==Extensions and late submissions== * No extensions or late submissions * If you don't submit, go ahead and sign up to an approved topic, and move on to the [[Motivation and emotion/Assessment/Topic|topic development exercise]] ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Select an appropriate, unique, specific motivation or emotion topic for the major project |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Be professional — communicate effectively | style="vertical-align:top;" | Communicate by signing up to a topic, submitting for approval, and asking clarifying questions |- | style="vertical-align:top;" | Be professional — display initiative and drive, and use organisation skills to plan and manage workload | style="vertical-align:top;" | Get organised by selecting a topic |- | style="vertical-align:top;" | Be a lifelong learner — evaluate and adopt new technology | style="vertical-align:top;" | Create a Wikiversity account and make at least 1 edit by signing up to a topic |} --> ==Instructions== Create a Wikiversity account and sign up to a pre-approved major project topic in the {{Motivation and emotion/Book}} or negotiate a new topic. This topic will be used for the [[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]. Ask any clarifying questions. ===[[Special:CreateAccount|Create a Wikiversity account]]=== * User names start with a capital letter * Can use real name, pseudonym, student number etc. ===Sign up to a topic=== '''Pre-approved topics''' are available via the {{Motivation and emotion/Book}} * How to sign up: ** Click "Edit" or "Edit source" ** Replace '''[[User:User Name|User Name]]''' alongside the topic of interest with your Wikiversity user name ** Click "Publish" <!-- * Conduct an initial literature search. If the topic is too broad or narrow, it may be difficult to satisfy the [[#Marking criteria|marking criteria]] --> * Pre-approved topics can be modified by emailing a revised title and sub-title to the [[Motivation and emotion/About/Staff|unit convener]] * You can change topics by moving your username to a topic without an author <!-- * More topics will be added, but students are also encouraged to propose topics --> ===New topics=== '''New or modified topics''' must be approved by the [[Motivation and emotion/About/Staff|unit convener]]. * Ways to get some ideas: ** [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Chat with the motivation and emotion book chapter topic generator] (Cogniti) ** Look through chapters written in [[Motivation and emotion/Book|previous years]] ** Check out [[Motivation and emotion/Book/Ideas for topics|similar projects]] on the internet * New topics must: ** '''Be unique''': Topic must not already be sufficiently covered by a [[Motivation and emotion/Book|previous motivation and motivation book chapter]]. Search before making a proposal. How does the proposed topic build on, or differ from, previous work? ** '''Align with the project theme''': Topic must be related to related to [[motivation]] or [[emotion]] and fit the overarching book theme which is to help people to understand and improve their motivational and emotional lives using psychological science ** '''Have appropriate scope''': If the topic is too narrow, or if there is a lack of psychological theory and research, it will be difficult to satisfy the [[#Marking criteria|marking criteria]] * To propose a new topic, email the [[Motivation and emotion/About/Staff|unit convener]] with these details: ** Title ** Sub-title (in the form of a question) — see [[Motivation and emotion/Book|examples]] ** Wikiversity user name ** Details of any related previous book chapter topics (check via this [[Motivation and emotion/Book|search box]]) ===Ask clarifying questions=== * What questions could you ask to help you to successfully tackle the major project? ==See also== * [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Selection| ]] bbh1t68hbcj2eb3nnrwkhromtvg5pmx 0 317777 2819702 2810130 2026-07-28T19:20:16Z ~2026-41928-17 3103894 new course 2819702 wikitext text/x-wiki {{law}} {| style="width:100%; margin-bottom: .6em; -moz-border-radius: 4px; text-align:center; border: 1px solid #50A6C2; background-color:#F0F8FF; padding: .6em .6em .6em .6em;" |<div style="font-size:200%;border:none;margin: 0;padding:.1em;color:#000">'''International Communication Law Learning Project'''</div><div style="top:+0.2em;font-size:150%">part of the [[School:Law|'''School of Law''']]</div> |} {{SearchWithPrefix}}'''[[Help:Resources by subject|Subject classification]]''': this is a [[Portal:Law|law learning projects]] resource. '''Class''': International Communication Law '''Professor''': Juan Carlos Riofrio [[File:International Communication Law.png|thumb|300x300px]]Welcome to the '''International Communication Law''' Learning Project, part of the [[School:Law|School of Law]]. This learning project aspires to establish the highest standards in the field of international communication law by fostering a comparative analysis of legal systems across different countries. By examining and contrasting the regulations, principles, and practices governing communication in diverse jurisdictions, the project aims to identify best practices and promote a deeper understanding of global legal standards. The ultimate goal of this project is to equip legal professionals with the knowledge necessary to navigate the complexities of cross-border communication, ensuring compliance with diverse legal systems while upholding fundamental principles of fairness and accuracy. == 2025 == [[Armenian Legal System]] [https://w.wiki/F6Yy France's Legal System] [[Italian Legal System|Italian Legal System]] [[Canadian Communications Law|Canadian Legal System]] [[Saudi Arabian Legal System|Saudi Arabian Legal System]] [[United Kingdom Legal System]] [[Puerto Rico's Legal System]] [[German Legal System]] [[Russian Legal System]] [[Jamaican Legal System]] [[Brazilian Legal System]] [[Communications Law in Austria]] [[Republic of Ireland Legal System]] [[Australia's Legal System]] == 2026 Spring == [[Communications Law in Malta]] [https://w.wiki/Hc5V The Kingdom of Morocco] [[Communications Law in China]] [[The Netherlands' Legal System|Communications Law in the Netherlands]] [[Communications Law in India]] [[Communications Law in Portugal]] [[Media Law in the Czech Republic]] [[Communications Law in Spain]] Communication Law in [[Communication Law in Spain|Spain]] [[Communications Law in Mexico]] == 2026 Fall == Communication Law in... [[Category:Law]] [[Category:Communication]] 8uxca1te15lp1n0oeegxmlci1butqq0 2819708 2819702 2026-07-28T20:04:25Z CommunicationLaw 2996029 /* 2026 Fall */ new entry 2819708 wikitext text/x-wiki {{law}} {| style="width:100%; margin-bottom: .6em; -moz-border-radius: 4px; text-align:center; border: 1px solid #50A6C2; background-color:#F0F8FF; padding: .6em .6em .6em .6em;" |<div style="font-size:200%;border:none;margin: 0;padding:.1em;color:#000">'''International Communication Law Learning Project'''</div><div style="top:+0.2em;font-size:150%">part of the [[School:Law|'''School of Law''']]</div> |} {{SearchWithPrefix}}'''[[Help:Resources by subject|Subject classification]]''': this is a [[Portal:Law|law learning projects]] resource. '''Class''': International Communication Law '''Professor''': Juan Carlos Riofrio [[File:International Communication Law.png|thumb|300x300px]]Welcome to the '''International Communication Law''' Learning Project, part of the [[School:Law|School of Law]]. This learning project aspires to establish the highest standards in the field of international communication law by fostering a comparative analysis of legal systems across different countries. By examining and contrasting the regulations, principles, and practices governing communication in diverse jurisdictions, the project aims to identify best practices and promote a deeper understanding of global legal standards. The ultimate goal of this project is to equip legal professionals with the knowledge necessary to navigate the complexities of cross-border communication, ensuring compliance with diverse legal systems while upholding fundamental principles of fairness and accuracy. == 2025 == [[Armenian Legal System]] [https://w.wiki/F6Yy France's Legal System] [[Italian Legal System|Italian Legal System]] [[Canadian Communications Law|Canadian Legal System]] [[Saudi Arabian Legal System|Saudi Arabian Legal System]] [[United Kingdom Legal System]] [[Puerto Rico's Legal System]] [[German Legal System]] [[Russian Legal System]] [[Jamaican Legal System]] [[Brazilian Legal System]] [[Communications Law in Austria]] [[Republic of Ireland Legal System]] [[Australia's Legal System]] == 2026 Spring == [[Communications Law in Malta]] [https://w.wiki/Hc5V The Kingdom of Morocco] [[Communications Law in China]] [[The Netherlands' Legal System|Communications Law in the Netherlands]] [[Communications Law in India]] [[Communications Law in Portugal]] [[Media Law in the Czech Republic]] [[Communications Law in Spain]] Communication Law in [[Communication Law in Spain|Spain]] [[Communications Law in Mexico]] == 2026 Fall == [[Communication Law in Ecuador]] [[Category:Law]] [[Category:Communication]] s3xautvuy2war3770pcwocp8745y5ao 0 319035 2819729 2811140 2026-07-29T00:27:37Z Unitfreak 695864 /* Slide 16 Notes */ 2819729 wikitext text/x-wiki {| class=table style="width:100%;" |- | {{Original research}} | [https://physwiki.eeyabo.net/index.php/Main_Page <small>Development <br/>Area</small>] |} [[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Foundations|The Foundations of Bully Metric]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> {{Gallery |width=700 |height=400 |File:Bully_Astronomical_Foundations.slide_1.svg | |File:Bully_Astronomical_Foundations.slide_2.svg | https://www.merriam-webster.com/dictionary/bully |File:Bully_Astronomical_Foundations.slide_3.svg | |File:Bully_Astronomical_Foundations.slide_4.svg | |File:Bully_Astronomical_Foundations.slide_5.svg | |File:Bully_Astronomical_Foundations.slide_6.svg | |File:Bully_Astronomical_Foundations.slide_7.svg | |File:Bully_Astronomical_Foundations.slide_8.svg | |File:Bully_Astronomical_Foundations.slide_9.svg | |File:Bully_Astronomical_Foundations.slide_10.svg | |File:Bully_Astronomical_Foundations.slide_11.svg | |File:Bully_Metric_Astronomical_Foundations.slide_12.svg | |File:Bully_Metric_Astronomical_Foundations.slide_13.svg | |File:Bully_Metric_Astronomical_Foundations.slide_14.svg | |File:Bully_Metric_Astronomical_Foundations.slide_15.svg | [[Bully_Foundations#Slide_15_Notes|Slide 15 Notes]] |File:Bully_Metric_Astronomical_Foundations.slide_16.svg | [[Bully_Foundations#Slide_16_Notes|Slide 16 Notes]] |File:Bully_Metric_Astronomical_Foundations.slide_17.svg | [[Bully_Metric_Astronomical_Coordinates|Slide 17 Notes (Bully Metric Astronomical Coordinates)]] |File:Bully_Metric_Astronomical_Foundations.slide_18.svg | |File:Bully_Metric_Astronomical_Foundations.slide_19.svg | |File:Bully_Metric_Astronomical_Foundations.slide_20.svg | }} == Slide 15 Notes == [[File:Hubbleconstants_color.png|thumb|left|800px|Selected estimated values of the Hubble constant, 2001-2019. Estimates in black represent calibrated distance ladder measurements, red represents early universe CMB/BAO measurements with ΛCDM parameters while blue are independent measurements.]] [[File:Look-back_time_by_redshift.png|thumb|right|400px|The lookback time of extragalactic observations by their redshift up to z = 20]] The exact value of the [https://lambda.gsfc.nasa.gov/education/graphic_history/hubb_const.html Hubble constant is unknown] ; consequently, the exact [https://en.wikipedia.org/wiki/Age_of_the_universe age of the Universe] can not be determined. By convention, timestamp 0000 0000 0000 is assumed to be the timestamp of the Big Bang. If the universe is 13.84 years old, then the hubble Constant is: [https://www.google.com/search?q=%281+sec%29+*+%281+megaparsec%29+%2F+%28%288+*+16%5E11+%2B+2+*+16%5E10%29+*+3055+s%29 70.66] km/s/Mpc. == Slide 16 Notes == Black hole Sagittarius A* appears to move (due to the orbit of the sun around the Milky-way) with a rate of six milli-arc-seconds per year. The motion is listed in Wikipedia as "[https://en.wikipedia.org/wiki/Sagittarius_A* approximately −2.70 mas per year for the right ascension and −5.6 mas per year for the declination]" or a vector addition total of 6.22 milli-arc-seconds per year. At this constant rate, Sagittarius A* would move six arc-seconds every thousand years. It would move six arc-minutes in 60 thousand years, or one arc-minute every 10 thousand years. It would move one degree every 600 thousand years, ten degrees every 6 million years, and 180 degrees every 108 million years. Unfortunately though, the motion is not constant, as the sun tends to drift up and down in an arm of the Milky Way as it orbits. Slide 16 uses an approximate value (106 million years) for the sun to orbit 180 degrees, not counting up and down drifting motion. lfzcup09gu065sioqz39rjgkuroihmt 0 323153 2819727 2819622 2026-07-29T00:15:24Z Jtneill 10242 /* Emotion */ + [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? 2819727 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}} # [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}} # [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}} # [[/Expectancy–value theory of educational motivation/]] - What is expectancy–value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}} # [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|U3183521}} # [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}} # [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|U3237996}} # [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Adaptive versus maladaptive self-reflection/]] – When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}} # [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}} # [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|User Name}} # [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}} # [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}} ==Motivation and emotion== # [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}} # [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}} # [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}} # [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}} # [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}} # [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book/2026]] ghlrfbaal27xujp6bl4kkvk6byt8mk0 2819728 2819727 2026-07-29T00:21:53Z Jtneill 10242 /* Emotion */ [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? 2819728 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}} # [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}} # [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}} # [[/Expectancy–value theory of educational motivation/]] - What is expectancy–value theory and how can it be applied to understand and enhance educational motivation? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|U3280251}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}} # [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|U3183521}} # [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}} # [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|U3237996}} # [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/The quiet ego and motivation/]] - How does a quiet ego balance self-interest with concern for others? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Adaptive versus maladaptive self-reflection/]] – When does self-reflection promote wellbeing and when does it contribute to psychological distress? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|U3285438}} # [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}} # [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mental health first aid and helping behaviour/]] - What motivates people to recognise, approach, and support someone with a mental health problem? {{ME-By|User Name}} # [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}} # [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Outdoor play and children's emotional well-being/]] - How does outdoor play influence children's emotional well-being? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}} ==Motivation and emotion== # [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}} # [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}} # [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}} # [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}} # [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}} # [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book/2026]] a1vlispiagzmrcvd2x0fq9h43w4ee18 0 325872 2819741 2819617 2026-07-29T02:08:54Z Jtneill 10242 [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Book chapter topic generator] 2819741 wikitext text/x-wiki __NOTOC__ <!-- __NOEDITSECTION__ --> <!-- Title - Box --> {{RoundBoxTop|theme=2}} <div style="text-align: center;"> <!-- Title --> {{title|[[Motivation and emotion/Book/2026|<big><big>Motivation and emotion</big></big>]]:}} <!-- Sub-title and year --> <div style="color: purple; font-size: large; font-weight: bold;"> Understanding and improving our motivational and emotional lives using psychological science (2026) </div> <!-- Initial development --> {{notice|<!-- Aiming for approximately ~150 topics - more coming.<br> -->These chapters<!-- will be--> are being developed by ~150 [[emerging scholar]]s from the [[w:University of Canberra|University of Canberra]], August to October, 2026, as part of their enrolment in [[motivation and emotion]].<!-- Editor: [[User:Jtneill|James Neill]].-->}} <!-- Training resources --> {| style="border:2px solid #616F7C;background-color:WhiteSmoke;padding:2px;width:80%;margin: 0 auto 1em auto;" |- |{{center top}} ;Training resources [[Motivation and emotion/Assessment/Topic|Topic development]] | [[Motivation and emotion/Assessment/Chapter|Book chapter]] [[Motivation and emotion/Lectures/Introduction|Lecture 01]] | [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] | [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] | [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] | [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Book chapter topic generator] {{center bottom}} |- |} </div> <!-- Initial message --> {| style="border:2px solid #616F7C;background-color:WhiteSmoke;padding:2px;width:80%;margin: 0 auto 1em auto;" |- |{{center top}} ;How to select a topic {{center bottom}} <!-- Pre-approved topics are listed below.<br> --> # [[Motivation and emotion/Wikiversity/Creating an account|Create an account and login]] # Look through available topics without a current author (indicated by "User Name") # Click "Edit" or "Edit source" # Replace "User Name" alongside the topic of choice with your Wikiversity user name # Publish the page # Check that your user name appears correctly; if not, fix it or [[Motivation and emotion/Help|get in touch]] # Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a new topic]] |- |} <!-- Drafting message --> <!-- These pages are undergoing a massive transformation.<br>~150 [[emerging scholar]]s who are studying [[Motivation and emotion|motivation and emotion]] are each authoring a resource about how psychological science can be used to understand and improve our lives.<br>Feel free to comment or contribute. --> <!-- Marking message --> <!--Most of the ~150 chapters have been submitted and are now undergoing expert review.<br>Feel free to continue improving and commenting.--> {{RoundBoxBottom}}<!-- {{Motivation and emotion/Book/Quality3}} --><noinclude>[[Category:Motivation and emotion|{{SUBPAGENAME}}]]</noinclude> 7933iiu0fliswoi9890tnv9nsd411t5 2819742 2819741 2026-07-29T02:10:45Z Jtneill 10242 2819742 wikitext text/x-wiki __NOTOC__ <!-- __NOEDITSECTION__ --> <!-- Title - Box --> {{RoundBoxTop|theme=2}} <div style="text-align: center;"> <!-- Title --> {{title|[[Motivation and emotion/Book/2026|<big><big>Motivation and emotion</big></big>]]:}} <!-- Sub-title and year --> <div style="color: purple; font-size: large; font-weight: bold;"> Understanding and improving our motivational and emotional lives using psychological science (2026) </div> <!-- Initial development --> {{notice|<!-- Aiming for approximately ~150 topics - more coming.<br> -->These chapters<!-- will be--> are being developed by ~150 [[emerging scholar]]s from the [[w:University of Canberra|University of Canberra]], August to October, 2026, as part of their enrolment in [[motivation and emotion]].<!-- Editor: [[User:Jtneill|James Neill]].-->}} <!-- Training resources --> {| style="border:2px solid #616F7C;background-color:WhiteSmoke;padding:2px;width:80%;margin: 0 auto 1em auto;" |- |{{center top}} ;Training resources [[Motivation and emotion/Assessment/Topic|Topic development]] | [[Motivation and emotion/Assessment/Chapter|Book chapter]] [[Motivation and emotion/Lectures/Introduction|Lecture 01]] | [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] | [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] | [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] | [https://cogniti.canberra.edu.au/agents/6a3caf85b2fed9a95789241a/chat?k=WfY1yx6AB7yF_9XqiyYENzUBNCGfV1e2FCBodCDD2xU Topic generator] {{center bottom}} |- |} </div> <!-- Initial message --> {| style="border:2px solid #616F7C;background-color:WhiteSmoke;padding:2px;width:80%;margin: 0 auto 1em auto;" |- |{{center top}} ;How to select a topic {{center bottom}} <!-- Pre-approved topics are listed below.<br> --> # [[Motivation and emotion/Wikiversity/Creating an account|Create an account and login]] # Look through available topics without a current author (indicated by "User Name") # Click "Edit" or "Edit source" # Replace "User Name" alongside the topic of choice with your Wikiversity user name # Publish the page # Check that your user name appears correctly; if not, fix it or [[Motivation and emotion/Help|get in touch]] # Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a new topic]] |- |} <!-- Drafting message --> <!-- These pages are undergoing a massive transformation.<br>~150 [[emerging scholar]]s who are studying [[Motivation and emotion|motivation and emotion]] are each authoring a resource about how psychological science can be used to understand and improve our lives.<br>Feel free to comment or contribute. --> <!-- Marking message --> <!--Most of the ~150 chapters have been submitted and are now undergoing expert review.<br>Feel free to continue improving and commenting.--> {{RoundBoxBottom}}<!-- {{Motivation and emotion/Book/Quality3}} --><noinclude>[[Category:Motivation and emotion|{{SUBPAGENAME}}]]</noinclude> re1w740ia9hvjmrktszq5hidumxwybf 0 326060 2819641 2818623 2026-07-28T11:59:33Z Jeanne Noiraud 1366702 /* 4. Writing */ 2819641 wikitext text/x-wiki == Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) == == Acknowledgements == The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. === Contributors === {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling, article writing |- |Amélie E. Pereira |Laboratoire DICEN IDF |https://orcid.org/0009-0005-5928-5586 |Meta-data enrichement, article writing |- |Finn Nielsen |Technical University of Denmark |https://orcid.org/0000-0001-6128-3356 |Data visualisation |} Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review == Introduction == Just sustainability transition refers to the process of shifting towards sustainable practices in a way that is equitable and inclusive. It includes dimensions of procedural, recognition, distributive and reparative justice and the concept is related to climate justice, environmental justice and energy justice<ref>{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}</ref><ref>{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}</ref>. The study of sustainability transitions in social sciences requires dynamic and adaptive research synthesis methods. Sustainability transitions involve complex, multi-level processes influenced by technological, economic, social, and policy factors<ref name=":15">{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}</ref><ref name=":16">{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}</ref><ref name=":17">{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research & Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}</ref>. Given the rapidly evolving nature of sustainability-related research, static literature reviews often become outdated, limiting their usefulness for policymakers, scholars, and practitioners. A living literature review – continuously updated with new findings – ensures that emerging insights, case studies, and theoretical developments are integrated cumulatively into the knowledge base. Developing such review will answer the call for more evidence-based practices in management sciences<ref>{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning & Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}</ref><ref>Pfeffer, J., & Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. </ref>. Our project assesses the potential of Wikidata to build living review workflow on sustainability transition. We address three issues encountered by scientists: information overload, knowledge synthesis and results dissemination. === The problem of academic information overload === Global scientific output is quickly growing<ref>{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}</ref><ref name=":23">{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}</ref><ref>{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}</ref>, pushed by the “publish or perish” model incentivizing researchers to increase the quantity of research outputs. Researchers are subject to information overload as the number of publications to read is beyond what a human brain can handle, they are expected to produce high-quality research under an increasing time pressure<ref name=":23" />. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge<ref>{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}</ref>. “Wikifying science” may in this context contribute to facilitating researcher’s work while preserving scientific quality. That is why in this project, we aim to build a searchable academic publication database with enriched meta-data that will allow scholars to navigate the existing publications corpus related to just sustainability transition more easily. === The problem of knowledge synthesis === The volume of academic production is rendering knowledge synthesis difficult. Scholars have thus called for making literature reviews cumulative and updatable<ref>{{Citation|title=Day 2 {{!}} Arnaud Vaganay: Reproducible Literature Reviews|url=https://www.youtube.com/watch?v=Nspd_1cx9kc|date=2017-10-19|accessdate=2026-06-23|last=Berkeley Initiative for Transparency in the Social Sciences (BITSS)}}</ref><ref>{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: "Perpetual Reviews"|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}</ref> and for shifting from static text format publications to dynamic knowledge mapping<ref name=":11">{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}</ref>. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : <ref>{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}</ref>,<ref>{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}</ref>,<ref name=":18">{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}</ref>). While such reviews method exist for quantitative research producing standardized results, they are not adapted to synthetize social science studies on sustainability transitions that involve diverse methodologies and various disciplinary perspectives. The goal of the project is to propose a demonstration of a living review method for social science findings on just sustainability transition, relying on the collaborative model and tools of Wikimedia projects notably Wikidata, Wikiversity and Wikipedia. === The problem of scientific results dissemination === There is urgent need to disseminate knowledge on impactful topics like sustainability transition while proprietary publication models, disinformation and censorship (e.g. US) is threatening access to free and reliable knowledge. In parallel, social scientists struggle to make their work impactful<ref>{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}</ref>. Wikipedia is a key knowledge dissemination platform widely used by students<ref>{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}</ref> and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature<ref>{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}</ref> and that some scholars cite directly Wikipedia<ref>{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}</ref>. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives<ref>{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}</ref>,<ref>{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}</ref>, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow<ref name=":10">Taraborelli, D., Mietchen, D., Alevizou, P., & Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. <nowiki>http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf</nowiki> </ref>. In addition, expert participation is not immune to the gender gap<ref name=":10" />. Because of gender segregation in disciplines<ref>{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}</ref>, this may be detrimental to the content coverage on “female” topics<ref>{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}</ref>, notably for social science in which women are more present. Our project proposes to improve expert contribution by making wikimedia projects (notably wikidata) useful tools that can facilitate research work, in addition to a key knowledge dissemination platform that is not country or institution-dependent. We propose to approach Wikimedia projects as a powerful (and free) knowledge management infrastructure that researchers could use. The Wikimedia ecosystem offers solutions that have strong potential to put open science principles into practices, including [[wikipedia:FAIR_data|FAIR]] principles and [[wikipedia:Linked_data#Linked_open_data|linked open data]]. == Toward a living review on just sustainability transition == === Just sustainability transition === Just sustainability transition transition is "a fair and equitable process of moving towards a post-carbon society"<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. Developping living reviews seem particularly relevant for the just transition literature: first, modeling knowledge and building graphs allows to take into account the complexity of sustainability transitions which involve multiple levels of analysis<ref name=":15" /><ref name=":16" /><ref name=":17" /> and fragmented results coming from various disciplines<ref name=":20">{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. Then, making literature reviews "living" would allow researchers to be less subject to information overload through a more systematic accumulation of knowledge. Finally, conducting this review with an open science philosophy aswers the challenge of knowledge dissemination, which is crucial in a context of socio-ecological emergency when decision-makers need to rapidely access reliable information on possible sustainability transition trajectories. === Living reviews === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1" /><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. Literature review methods are currently evolving with new technological possibilities. Generative artificial intelligence such as ChatGPT are expected to have a strong influence on literature review activities<ref name=":12">{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref name=":12" />, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but not yet integrated into tested and validated methodologies<ref name=":13">{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. Human validation stays notably necessary<ref>{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}</ref>,<ref name=":13" />. While AI can appear as a solution for scaling literature reviews, we are in the present project exploring another possible scenario which is to use more crowdsourcing in the literature review process. === Wikimedia projects === Wikipedia is a successfull example of large-scaled crowdsourcing of reliable knowledge synthesis. That is why this project proposes to explore the potential of the Wikimedia ecosystem for conducting living reviews. Since Wikipedia does aim to host original research<ref>{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&oldid=1360514388|journal=Wikipedia|language=en}}</ref>, we are working on two sister projects : Wikidata and Wikiversity. As its name suggests (wiki and data), [[wikipedia:Wikidata|Wikidata]] is, like Wikipedia, a collaborative project based on wiki editing technology; however, unlike the online encyclopaedia, it contains data rather than articles. Whilst there is a separate Wikipedia for each language version, there is only one Wikidata, containing items organised using unique identifiers. What matters when organising information in Wikidata is not the words used to define each concept, but the concept itself. Wikidata is a [[Databases|database]], more precisely a knowledge base, and a "collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation".<ref>{{Cite news|last=Chalabi|first=Mona|date=April 26, 2013|title=Welcome to Wikidata! Now what?|url=https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|access-date=October 2, 2021|archive-date=2 October 2021|archive-url=https://web.archive.org/web/20211002152920/https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|url-status=live}}</ref><ref>{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&oldid=1360462340|journal=Wikipedia|language=en}}</ref> "A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> Such graphs have a strong potential to conduct knowledge synthesis<ref name=":11" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref><ref name=":18" />. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge<ref name=":14">{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}</ref>. In complement to using Wikidata to model knowledge, we decided to use Wikiversity to report and write our research results. [[wikipedia:Wikiversity|Wikiversity]] is another Wikimedia project hosting pedagogical content, original research, and even a publishing house ([[WikiJournal|WikiJournals]])<ref>{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&oldid=1358552930|journal=Wikipedia|language=en}}</ref>. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. Our research question is : '''How can Wikimedia projects contribute to building a collaborative living review on just sustainability transition ?''' In this project, we aim to test 4 hypothesis : ● '''Hypothesis 1:''' Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations. ● '''Hypothesis 2:''' Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (e.g. conceptual typologies, cause-effect chains…). ● '''Hypothesis 3:''' SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs. ● '''Hypothesis 4''': Wikimedia or Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links (following the ideal of linked open data). We also have 2 assumptions : ● '''Assumption 1:''' Wikimedia projects have to be integrated into validated scientific protocols in order to be a valuable research tool. ● '''Assumption 2:''' Wikimedia project contribution has to be made interoperable with tools, methods and data types already used by researchers. == Methodology == Our study rely on a meta-review, that is a review of existing literature reviews. Data presented in literature reviews are usually presented as tables or diagrams, and sometimes provided as supplementary materials in publications. However, these data are not made interoperable and are not used to update prior literature reviews. Our goal was to synthesize results of previous literature reviews by making their findings compatible with linked open data and open science standards using Wikidata, Wikiversity, and other open-science infrastructures. The first step was to build and enrich the bibliographic metadata of a corpus of articles we selected into Wikidata. The second step was to model the content of these articles in Wikidata (e.g. typologies, causes-effects relationships...). The third step was to experiment relevant visualization of this content (e.g. causes-effects graphs). The las step was to write our report on a Wikiversity page, including links to our knowledge graph, following a linked open data philosophy. == 1. Building an academic corpus and enriching bibliographic metadata == The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations). To do so we imported academic references into Wikidata, and explored the advantages of constituting a scholarly corpus on Wikidata in comparison (or in complementarity) to existing tools used by researchers such as reference management softwares and knowledge management softwares. Reference management software (Zenodo, Mendeley…) are used to collect scientific item metadata and integrate them into academic writing. They can also be used to analyze and annotate academic articles and can include export functions making the data interoperable with other analysis tools. Knowledge management software (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…) are used by some researchers to organize their ideas. To build and enrich our academic corpus on Wikidata, we searched existing databases, selected the sample of articles we wanted to study, imported these articles metadata into Wikidata, enriched these metadata and finally reflected on the advantages and limitations of Wikidata to build a rich academic corpus. === Database search === Doing a systematic review on all aspects of just transition would have resulted in too many articles to review. We thus decided to first explore one aspect of justice : procedural justice. Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref> (e.g. the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />). Procedural justice can include issues of community and citizen participation in decision making, their political representation, their consultation, or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities)<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For our search, we selected keywords related to procedural justice (procedural justice OR procedural fairness OR democracy OR participation OR participatory) and keywords related to sustainability transition (sustainability OR energy OR climate) AND (transition OR transitions). We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article selection === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were : * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies/initiatives/cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors * Including justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper The files with the lists of included and excluded articles are available on the archive plateform Zenodo : https://zenodo.org/records/20749974 === Importing selected articles into Wikidata === Before importing the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in Wikidata. Next, we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through metadata enrichement === Metadatas are data describing other data. The metadata of academic items usually include title, author, publication outlet, publication date, pages, DOI, URL... and can be structured following specific standards (e.g. [[wikipedia:Dublin_Core|Dublin Core]]). In academic databases such as WOS or OpenAlex, the only metadata available regarding the content of an academic article are the abstract and sometimes keywords. However, researchers conducting literature reviews need more precise informations. An important part of literature review work can thus be about describing what the articles are about. For example, describing industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt) (e.g. <ref name=":5" />). By metadata enrichment, we mean completing metadata to include additional information about the content of an academic piece. In Wikidata, each type of information is added using a specific property. A property is the edge that links two entities in the Wikidata knowledge graph. We selected three Wikidata properties to describe the content of our selected articles : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe the geographical scope of the study. We also worked on adding {{Wikidata entity link|P50}}. ==== Adding {{Wikidata entity link|P921}} ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |}Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Adding {{Wikidata entity link|P8363}} ==== The sample of article we selected included only litterature reviews, but we wanted to describ what kind of literature review it was. We first read abstracts to identify all the different types of litterature reviews present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved some of these method items using the methodological references cited in the reviewed papers. For example, we added a statement saying that {{Wikidata entity link|Q101116078}} can have {{Wikidata entity link|Q653137}} as {{Wikidata entity link|P13391}}<ref>{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information & Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}</ref>. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |theory-driven evaluation used in evaluating social programmes |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |study of the statistical properties of combinations of studies from a meta-analytic dataset |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}For each article, we added the {{Wikidata entity link|P8363}} based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Adding {{Wikidata entity link|P6153}} ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Adding {{Wikidata entity link|P50}} ==== When scholarly metadata are imported into Wikidata, the name of authors are stored as a chain of characters and linked to the property {{Wikidata entity link|P2093}}. The property {{Wikidata entity link|P50}} allows to make a link with a Wikidata item representing the author. This avoids the problem of homonym authors by linking a unique identifyer to authors in Wikidata and linking this identifier to existing ones such as ORCID. We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers by categorizing scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages. ==== Adding {{Wikidata entity link|P6977}} ==== We explored the possibility to model review networks, that is linking review paper to reviewed papers (and not broadly cited papers). We selected one of the paper which had the most detailed list of reviewed papers ({{Wikidata entity link|Q114306483}}), then we added this list in Wikidata using Open refine. Then we snowballed this process for the reviewed papers which were also litterature reviews. === Advantages and limitations of Wikidata to build a rich living academic corpus === To share the result of our work, we exported the dataset we build on Wikidata and shared it on the open archive Zenodo : https://doi.org/10.5281/zenodo.20749973. The data is also available directly in Wikidata. The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations)'''.''' The sections below discuss the advantages and limitations of wikidata regarding this hypothesis. ==== Advantages of Wikidata ==== Key advantages of Wikidata are its flexible and collaborative nature as well as its interoperability. Wikidata ontology (that is how the data are structured) is collaboratively defined and properties can be added if relevant (after validation by the community). Compared to global databases like WOS or OpenAlex, Wikidata allows anyone to enter more metadata about each academic articles. Another notable advantage is that Wikidata items can be used as an interoperable [[wikipedia:Controlled_vocabulary|controlled vocabulary]]. For example, when we stated that the {{Wikidata entity link|P921}} of the article {{Wikidata entity link|Q114306483}} was {{Wikidata entity link|Q795757}}, "energy transition" was not just a word but a concept with its unique identifyer, linked to identifiers in other databases such as the Google Knowledge Graph ID or BNCF Thesaurus ID. Wikidata's collaborative nature is here adain an advantage. Contrary to institutional thesaurus, Wikidata allows anyone to add new concepts. This is particularly interesting as existing controlled vocabularies rarely reflect the degree of precision that researchers need in their work. The multilingual nature of Wikidata was also a strengh, some Wikidata contributors added labels for the concepts we used into different languages (For example, contributors added labels for {{Wikidata entity link|Q14944319}} in Armenian and Slovenian, languages we do not speak at all). ==== Limitations of Wikidata ==== The limitations of Wikidata is that it is not yet well integrated with the tools researchers use to do literature reviews (kowledge management softwares and bibliographic databases). Compared to reference management softwares (Zenodo, Mendeley…) and knowledge management softwares (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…), Wikidata is too general and does not allow to work on full texts. References and knowledge management softwares allow researcher to build their own specialised knowledge base, by taking notes and highlighting the content of the full texts. Wikidata is not connected to this process and there is a missing tool to facilitate the construction of graphs from the qualitative analysis of texts. In addition, when one is working on a specific corpus of item in Wikidata, it is also difficult to keep track of this corpus. We linked each academic item we were working on to our research project by adding a statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}, but it was still relatively difficult to "filter" the part of the knowledge graph we were working on. Compared to bilbiographic catalogues (OpenAlex, Web Of Science, GoTriple...), Wikidata will never be as exhaustive and do not offer user-friendly search functions. Since 2014, an important amount of bibliographic data was imported in Wikidata with the project [[d:Wikidata:WikiCite|Wikicite]]. At the time of its creation, Wikicite was adressing the issue of closed bibliographic data and was trying to make these data open, many academic items were imported automatically in Wikidata through scraping. This practice was abandoned because the large amont of bibliographic data congested queries on Wikidata (this led to the decision to split the Wikidata graph between academic and non academic entities), and because new open science initiatives, notably OpenAlex (2022), are now taking on the task of creating a exhaustive catalogues of all scholarly production. In this context, the community has to rethink the purpose of Wikidata regarding bilbiographic data. ==== Future possbilities ==== A solution to the limitations would be to developp the role of Wikidata as a link between other tools of the open science ecosystem. For example, developping and maintaining plugins or extensions for specialised softwares like Zotero, Wikibase, and Omeka could connect Wikidata with more specialised graphs. Such extensions could help building local graphs by allowing the reuse of wikidata item (eg. autocompletion), but also help contributing to Wikidata thanks to export features. (Example :<ref>{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}</ref>) Wikidata could then be an intermediary between locally curated corpus and more exhaustive bilbiographic catalogues such as OpenAlex. For example, Wikidata items could be used to tag articles in a more precise way instead of using keywords and sharing enriched corpus in Wikidata could help to train more precise taging algorythms. == 2.Modelling the content of litterature reviews == The goal of this step was to test '''Hypothesis 2''' (Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference) by modelling the content of our selected articles into Wikidata. [[wikipedia:Knowledge_modeling|Knowledge modelling]] is the process of making a machine readable model of knowledge. As we have a background in social sciences, we felt the need to question the relationship between this process and other methodologies such as concept mapping, thematic networks and causal networks. We present these methodologies before describing our current knowledge modelling experimentations. === Concept mapping, thematic networks and causal networks === This section presents social science methodology that presents similarities with knowledge modelling. ==== Concept maps ==== [[File:Conceptual_Diagram_-_Example.svg|link=https://en.wikipedia.org/wiki/File:Conceptual_Diagram_-_Example.svg|thumb|Example conceptual diagram|251x251px]]Concept maps are ''concepts'' (boxes) and ''propositions'' (arrow indicating the relationship between two boxes)<ref name=":19">Cañas, Alberto J., et al. "CmapTools: A knowledge modeling and sharing environment." (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf</ref>. Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]<ref name=":19" />, Altas.Ti "network" feature...). The "box and arrow" logic is similar to how knowledge is modelled on Wikidata : the equivalent of concepts is ''item'' and the equivalent of propositions are ''statements''. The difference between a softwares like Cmap and Wikidata is the underlying format of the data. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|447x447px|Structure of a thematic network (Source: based on Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. [[File:Adoption_CLD.svg|link=https://en.wikipedia.org/wiki/File:Adoption_CLD.svg|thumb|421x421px|Causal loop diagram of ''Adoption'' model, used to demonstrate systems dynamics]] ==== Causal diagrams ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref name=":21">{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> After identifying themes, qualitative researchers are expected to theorize the "arrows" between themes<ref name=":21" />R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. === Knowledge modelling in Wikidata === This section presents our knowledge modelling experimentation in Wikidata. ==== Conceptual modelling ==== Our first step was to reflect on what is a "concept" and what kind of wikidata properties could be used to model concepts in Wikidata. Scholars in management have called for more rigorous ways to define concepts<ref name=":22" /> and modelling concepts in Wikidata could help to build less ambiguous concepts. Concept definition encompass various aspects such as the nature of the phenomenon, its characteristics, the links with prototypical cases or examples, the contrast with other concepts, the links with causes and consequences...<ref name=":22">{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}</ref>, and scholars have advised to take insight from philosophy to work on concepts<ref>{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}</ref>. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref>. We summarize these approaches below and examine which wikidata properties exist to represent them. *Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... *Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. (The closer a phenomenon is to the prototype, the more likely it belong to the category). Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}} (see discussion here https://www.wikidata.org/wiki/Help:Basic_membership_properties). *Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}, {{P|1478}}, {{P|P9353}} (see discussions here : https://www.wikidata.org/wiki/Help:Modeling_causes/en). *Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can use it before really knowing what its properties will be. *Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Testing concept modelling on {{Wikidata entity link|Q14944319}} ==== To test concept modelling, we started by experimenting with the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic : *{{Wikidata entity link|Q137901202}} *{{Wikidata entity link|Q137901196}} *{{Wikidata entity link|Q137901182}} *{{Wikidata entity link|Q136447761}} *{{Wikidata entity link|Q129652515}} *{{Wikidata entity link|Q114306483}} We read each paper and used them as source in manually entered statements in the item {{Wikidata entity link|Q14944319}}. For example, Droubi et. Al stated "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}} and {{Wikidata entity link|Q3249551}}, using the paper as reference (see screenshot below). The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&oldid=2495982191. [[File:Wikidata statement- energy democracy is an instance of ideal.png|915x915px|border]] We listed the difficulties encountered as we worked and we also asked the Wikidata community to give us feedback on our modelling on the item discussion page (https://www.wikidata.org/wiki/Talk:Q14944319). These issues were related to contradictions, precision, concision and ontology. We discuss each issue and draft recommandations to refine our modelling process. ===== Contradictions ===== Wikidata contributor's feedback highlighted some apparent contradictions (The values in "does not have effect" seems contrary to what is listed in "has goal".) We would however argue this is not a problem because "statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata"<ref>{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}</ref>. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref>.Besides, in the case of goals versus effects statements, it is not contradictory because one can have a goal and fail to achieve it. In the case of energy democracy, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing<ref name=":20" />. Recommandations : Contradictions are allowed in Wikidata. ===== Precision ===== We noted that conceptual modelling requires an important degree of formalization and precision. This is a key advantage of Wikidata to be able to create links toward precise concepts which have their own identifiers. For example, we were able to create statements about specific laws and their unique identifiers in legal databases (e.g. {{Wikidata entity link|Q139764294}} and its identifier in the EUR-Lex database). However, the sources we are working with are not always precise enough and when concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, {{Wikidata entity link|Q12888920}}... We can see here that academic texts are using natural language and thus are using ambiguous terms. As a result, we received feedbacks regarding a lack of precision in our statements (too many and too vague statements). In addition to the ambiguity of sources, a reason why we ended up with very general statements is because we avoided the creation of new Wikidata items. While following this implicit rule allowed us to focus on the most notable concepts, creating new items could also help make the statements more precise. Recommandations : If a concept is ambiguous it should not be included in the modelled statements. Create more precise relevant concepts if they do not exist in Wikidata. [[File:Wikidata visualisation screenshot of subclasses relationships including the item political concept.png|thumb|298x298px|Subclass relationships between "concept" and "political concept".]] ===== Concision ===== Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were "in the tree of another value". The rule we take from this feeback is a need of logical simplification. Two examples illustrate possible logical simplification : *We stated that {{Wikidata entity link|Q14944319}} was an {{Wikidata entity link|P31}} {{Wikidata entity link|Q33104069}} and an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}. But in that case, it is not necessary to state that it is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}, because {{Wikidata entity link|Q33104069}} is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q131362181}}, which is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q151885}} (see diagram on the right). Here, we have to keep only the more precise item. This reasonning is based on the assumption that {{Wikidata entity link|P279}} is transitive. It seems this reasonning could be generalized (we opened a discussion about this here https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning/Use_cases#Parcimonious_statement_constraints_based_on_subclass_of_(P279)_and_part_of_(P361)_transitivity<nowiki/>) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using "instance of" as transversal property and checking "instance of or subclass of"). *We stated that {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P2670}} {{Wikidata entity link|Q15991216}} and {{Wikidata entity link|Q113514984}}. But if we consider that {{Wikidata entity link|Q15991216}} is a {{Wikidata entity link|P279}} of {{Wikidata entity link|Q113514984}}, then the inclusion of {{Wikidata entity link|Q15991216}} is implied. Here we could keep only the broader item {{Wikidata entity link|Q113514984}}. But unlike the reasonning above, we could not identify a way to generalize this reasonning. For example, {{Wikidata entity link|Q113514984}} can be a subclass of {{Wikidata entity link|Q43229}}, but using the former item is more informative than using the later. For the property {{Wikidata entity link|P2670}}, using a value that is too precise results in too many statements and using a value that is too general results in too trivial statements. Choosing the degree of precision is a problem similar to choosing whether or not to create a new item. Recommandations : When possible, use logical simplification to make statements more parcimonious. The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize redundancies. ===== Quantification ===== Modelling quantitative statements was challenging. We mostly skipped those but made an attempt for one case : the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources"<ref>{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research & Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}</ref>. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for "final energy consumption". There are such properties in Wikidata : for example, for renewable energy subsidies, there is {{Wikidata entity link|P6826}}. This type of property is notably useful to display information about an item (e.g. the population of a town) on Wikipedia through dynamic infoboxes that are updated with Wikidata information. But the current informal rule on Wikidata property creation is to prefer the use of general properties and avoid the creation of specific properties. Therefore creating properties for every quantifiable characteristic may not be possible. An alternative way to model quantity is to create a specific item to represent what is quantified and use other properties to represent quantities (see example below). [[File:Wikidata screenshot of percentage modelling.png|border|783x783px]] Despite this possibility, since quantitative data are usually stored elsewhere in more convenient formats, it may not be always relevant to include quantitative data into Wikidata. Instead, it might be more useful to link the item that is quantified to an external dataset about it. This can be done using the property {{Wikidata entity link|P1325}}. Recommandation : Use existing properties for the quantity if it exists. If it does not exist, create an item representing what is measured, link it toward an external datasets and add quantitative statements if relevant. ===== Ontology ===== Ultimately, several of our difficulties were linked to some ontology challenges reflecting the complexity of the concept we were working on. According to the litterature, {{Wikidata entity link|Q14944319}} refer to multiple types of entities. It represents a concept, an ideal, a process and an outcome. We reflected this with multiple {{Wikidata entity link|P31}} statements, but encountered struggles to model differences between ''process versus outcomes'', ''ideal versus reality'', ''phenomenon versus theory'' and ''discourses versus practices''. The wikidata community recommands having different items for ''process and outcome'' (criterion "{{Wikidata entity link|Q127270577}}"). For example, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork). However, this distinction is less straightforward for social processes which are are ongoing and evolving without a clear ending. Another problem was to separate ''ideal versus reality,'' we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes attempting to reach it. It was also difficult to separate ''phenomenon versus theory,'' for example, we did not manage to model the idea that the literature on energy democracy is fragmented. Finally, we also would have needed more distinctions between ''discourses and practices.'' Our sources suggested that energy democracy discourses and practices may have different causes, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movments promoting it, but this was not a very precise way to model this idea. Our first attempt to multiple "natures" of energy democracy into a single item is not satisfactory, and implementing the modelling recommandations we identified above may not be sufficient to build more parcimonious statements. Creating new items to reflect the different aspects of energy democracy may here be necessary. To do so we have to decide how to split the different natures {{Wikidata entity link|Q14944319}} and what will be the relationship between the resulting items. Here the definition of an {{Wikidata entity link|Q324254}} (formal representation) in Wikidata can quickly escalate into questionning {{Wikidata entity link|Q44325}} (metaphysical reflexion on the nature of things), and especially {{Wikidata entity link|Q1713511}} (the nature of the social world). Indeed, the relations between the different types of entities constituting the social worls are not simple nor consensual among philosophers of social science. For example, {{Wikidata entity link|Q15080858}} posits that different things have different ways of being (modes of reality). They propose to classify entities in four categories : material entities (that can exist independently of humans), conceptual entities (concepts, discourses, ideas, meaning…), artefactual entities (human-made and combining conceptual and material elements) and social entities (that depends on human activity to exist)<ref>Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.</ref>. A complex concept like {{Wikidata entity link|Q14944319}} is concerned with all these types of entities. The energy system include many material entities such as oil fields, the sun, seas, trees... and artefacts such as energy production unit, power lines, home appliances, trucks... (Including conceptual entities such as the name of these artefacts or the knowledge to make them function.). There are the social entities in which they are encompassed (the enregy sectors, energy businesses, energy policies...). There are conceptual entities like normative/political discourses discussing how these artefact and social system should work and there are conceptual entities in the academic sphere building theories about how all this works or should work. In theory, such ontology can be implemented in Wikidata as it "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8" /> Current Wikidata ontology is structured with dischotomies such as {{Wikidata entity link|Q7048977}}/{{Wikidata entity link|Q4406616}}, {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}, {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q23958946}}, {{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}, {{Wikidata entity link|Q30241068}}/{{Wikidata entity link|Q3799040}}... A quick search seem to show that entities related to the social reality can quickly fall into heterogeneous and sometime contradictory root classes (see table below). {| class="wikitable" |+ !Item !Existing root class entity |- |{{Wikidata entity link|Q190539}} |{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q67518978}} |- |{{Wikidata entity link|Q34770}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q9081}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q43229}} |{{Wikidata entity link|Q99527517}}{{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q103940464}} |- |{{Wikidata entity link|Q49773}} |{{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}} {{Wikidata entity link|Q99527517}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q12705}} |{{Wikidata entity link|Q4406616}}/{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}} {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q132907471}} |} === Future steps === Current {{Wikidata entity link|Q3882785}} in Wikidata do not seem very informative to classify the aspects of our concept in a top-down manner. Our next steps will thus be to : * Export the existing model of the concept to keep track of each modelling steps (files and graphs screenshots) *Explore if existing standards offer more relevant classes to categorize social entities (we opened the discussion about social ontology with the wikidata community here https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Ontology#Social_ontology). **CRMSoc : https://cidoc-crm.org/crmsoc/fm_releases **Criteria for choosing an ontology = choose something that is used and maintained by a community (colleague feedback) *Create new items to reflect the different facets of {{Wikidata entity link|Q14944319}} : the social movement advocating for it, the political concept theorizing it, the concrete initiatives implementing it, the litterature theorizing it, etc... based on a new reading of our sample of article and aligned with existing ontologies. *Compare grounded theory/CAQDAS with knowledge modelling methods, test annotation softwares : https://inception-project.github.io/ ; https://pacte.crim.ca/index_en.html *Read articles again and model more precise statements, including quotation of the original text in the reference field. *Write a more precise research question to orient modelling : What are the main concepts of just transition, their definition and their relationships ? * == 3. Data visualisation == The goal of this step is to test '''Hypothesis 3''' (SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs). At this stage we started to identify visualization use cases we would like to test, they are listed below. === Filter statements === *Visualize only statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH *Visualize only items which are part of the present project (all the academic items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Properties visualisation === *Visualise the "tree" of a property used in an item : Wikidata graph builder seem to be the most user friendly, robust and versatile tool to visualise a graph of a single property (https://angryloki.github.io) === Concepts visualisation === *Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request "topic in context") *Map the statements that two items have in common to make comparisons. === Mapping sources consensus === *Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible. == 4. Writing == The goal of this step is to test '''Hypothesis 4''' (Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links). * To do idea : write a Wikiversity page for each central concept (e.g. energy democracy, energy community, energy citizenship...) === Advantages of Wikiversity === Writing on a Wikiversity page offers some advantages to implement the principles of open linked data in text format. We could cite academic items using their Wikidata QID to generate the citations below (but if we use an URL to cite a paper, it does not automatically check if there is an existing QID for it), and also link toward Wikidata entities using a template ([[Template:Wikidata entity link|Wikidata entity link]]). Although we did not translate the present page for now, Wikiversity offers the possibility to translate a page in multiple languages (an interesting possibility in the context of the The FAIRisation of scholarly communication<ref>{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}</ref>). The possibility to view the page history provide an exhaustive versionning of a paper, which constitute interesting data documenting the academic writing process (at first we did not comment the nature of our modifications but after thinking about this reuse possibility, we described it with more details). The contribution statistics based on the history also offers a new way to track author's contributions to a paper in a transparent way. Some researchers experimented relationships between the wikimedia community and researchers by making researchers review wikipedia articles<ref>{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}</ref>. === The issue of text interoperability === A key issue we are encountering is the question of the interoperability of texts. While the interoperability of data is starting to be well discussed in the open science community, the interoperability of texts do not seem to benefit from the same level of discussion. We encountered several interoperability issues regarding our writing. First, copying texts written on a word processor software (e.g. microsoft word) into a wiki page (or the other way around) is relatively seamless in terms of formatting, but the management of references is lacking. Reformatting references is very time consuming and a real barrier for text interoperability in academic context : it is difficult to copy text from an academic publication into a wiki text, and difficult to turn a wiki text into a publication (the only possibility is to export the page as a PDF and you may be reading this article in this format). There are also uncertaineties regarding how to reuse texts published under creative common licences. Academic texts published under CC-BY-SA licences can in theory be remixed and reused. But academia does not have established practices regarding how this can be done. If we want to reuse a whole page, should we put it in quotation marks and simply cite the paper ? Should the original authors be listed as co-authors ? To what extent academic publisher will accept to publish "remixed" texts while they usually require that publications contain mainly unpublished content ? The norms of what is appropriate remix and reuse practices in academia has yet to be decided... and this could be an interesting conversation to have in the open science community. == Discussion == Our goal was to assess the potential of Wikidata and Wikiversity to build a living literature review method and tackle issues of information overload, knowledge synthesis and knowledge dissemination, following open science principles. By conducting a meta-review on just sustainability transition (that is a review of existing literature reviews), and experimenting with existing technical solutions, we were able to identify the strenghs and limitations of the Wikidata ecosystem. First Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations (Hypothesis 1 is verified). However, the technical barriers to do so are still high and the Wikidata database is too general to allow a community to work on a specific curated corpus of scientific items. Secondly, Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (Hypothesis 2 is verified). However, this requires developping new methodological standards regarding what is a rigorous modelling in social science, as well as building specific {{Wikidata entity link|Q324254}}. An unexpected finding is that the formalization necessary for knowledge modelling in wikidata invites to formalize the reasonning behind modelling choices, a process that is rarely done in qualitative analysis even though such reasonning are likely present. The modelling process also responds to calls in management sciences to engage in more systematic categorizing to avoid semantic confusion<ref>{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}</ref>. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to "navigate" it (Hypothesis 3 is partially verified). SPARQL-based queries are powerful but they require technical knowledge, especially now that the split between academic and non-academic items in Wikidata requires to write federated queries. To be able to "navigate" scientific corpora and scientific knowledge graphs on Wikidata, more user-friendly tools would be needed. Finally, we did see advantaged in using Wikiversity pages to write collaboratively in text format augmented by hypertext links, but there are still important technical and instittional barriers (Hypothesis 3 is partially verified) : the interoperability of text is a key issue to reuse publications. The main issues we encountered were the management of references and uncertainty regarding the compatibilities between writing publicly on a wiki page and engaging in classical publication processes. A potential solution to the issues encountered could be to develop a specialised literature review software that would allow researchers to build living literature reviews including knowledge graphs and wiki pages. This kind of tool should be user-friendly and include the missing tools we identified. In addition, it could include more advanced analysis functions such as logical reasonning based on the knowledge graph<ref name=":9" />. === Future research === - Can we "reason" with Wikidata (See: https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning) : neurosymbolic AI ? == Funding == This project is funded by the [[m:Grants:Programs/Wikimedia_Research_&_Technology_Fund/Wikimedia_Research_Fund|Wikimedia Research Fund]], Grant ID: G-RS-2504-18935. The text of the initial research proposal is available here : https://doi.org/10.5281/zenodo.20760603. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} d4mwqjrzh3bnmycxik3ny07mvw5w7qi 2819648 2819641 2026-07-28T13:31:12Z Jeanne Noiraud 1366702 /* Quantification */ 2819648 wikitext text/x-wiki == Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) == == Acknowledgements == The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. === Contributors === {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling, article writing |- |Amélie E. Pereira |Laboratoire DICEN IDF |https://orcid.org/0009-0005-5928-5586 |Meta-data enrichement, article writing |- |Finn Nielsen |Technical University of Denmark |https://orcid.org/0000-0001-6128-3356 |Data visualisation |} Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review == Introduction == Just sustainability transition refers to the process of shifting towards sustainable practices in a way that is equitable and inclusive. It includes dimensions of procedural, recognition, distributive and reparative justice and the concept is related to climate justice, environmental justice and energy justice<ref>{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}</ref><ref>{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}</ref>. The study of sustainability transitions in social sciences requires dynamic and adaptive research synthesis methods. Sustainability transitions involve complex, multi-level processes influenced by technological, economic, social, and policy factors<ref name=":15">{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}</ref><ref name=":16">{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}</ref><ref name=":17">{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research & Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}</ref>. Given the rapidly evolving nature of sustainability-related research, static literature reviews often become outdated, limiting their usefulness for policymakers, scholars, and practitioners. A living literature review – continuously updated with new findings – ensures that emerging insights, case studies, and theoretical developments are integrated cumulatively into the knowledge base. Developing such review will answer the call for more evidence-based practices in management sciences<ref>{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning & Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}</ref><ref>Pfeffer, J., & Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. </ref>. Our project assesses the potential of Wikidata to build living review workflow on sustainability transition. We address three issues encountered by scientists: information overload, knowledge synthesis and results dissemination. === The problem of academic information overload === Global scientific output is quickly growing<ref>{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}</ref><ref name=":23">{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}</ref><ref>{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}</ref>, pushed by the “publish or perish” model incentivizing researchers to increase the quantity of research outputs. Researchers are subject to information overload as the number of publications to read is beyond what a human brain can handle, they are expected to produce high-quality research under an increasing time pressure<ref name=":23" />. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge<ref>{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}</ref>. “Wikifying science” may in this context contribute to facilitating researcher’s work while preserving scientific quality. That is why in this project, we aim to build a searchable academic publication database with enriched meta-data that will allow scholars to navigate the existing publications corpus related to just sustainability transition more easily. === The problem of knowledge synthesis === The volume of academic production is rendering knowledge synthesis difficult. Scholars have thus called for making literature reviews cumulative and updatable<ref>{{Citation|title=Day 2 {{!}} Arnaud Vaganay: Reproducible Literature Reviews|url=https://www.youtube.com/watch?v=Nspd_1cx9kc|date=2017-10-19|accessdate=2026-06-23|last=Berkeley Initiative for Transparency in the Social Sciences (BITSS)}}</ref><ref>{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: "Perpetual Reviews"|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}</ref> and for shifting from static text format publications to dynamic knowledge mapping<ref name=":11">{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}</ref>. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : <ref>{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}</ref>,<ref>{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}</ref>,<ref name=":18">{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}</ref>). While such reviews method exist for quantitative research producing standardized results, they are not adapted to synthetize social science studies on sustainability transitions that involve diverse methodologies and various disciplinary perspectives. The goal of the project is to propose a demonstration of a living review method for social science findings on just sustainability transition, relying on the collaborative model and tools of Wikimedia projects notably Wikidata, Wikiversity and Wikipedia. === The problem of scientific results dissemination === There is urgent need to disseminate knowledge on impactful topics like sustainability transition while proprietary publication models, disinformation and censorship (e.g. US) is threatening access to free and reliable knowledge. In parallel, social scientists struggle to make their work impactful<ref>{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}</ref>. Wikipedia is a key knowledge dissemination platform widely used by students<ref>{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}</ref> and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature<ref>{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}</ref> and that some scholars cite directly Wikipedia<ref>{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}</ref>. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives<ref>{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}</ref>,<ref>{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}</ref>, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow<ref name=":10">Taraborelli, D., Mietchen, D., Alevizou, P., & Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. <nowiki>http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf</nowiki> </ref>. In addition, expert participation is not immune to the gender gap<ref name=":10" />. Because of gender segregation in disciplines<ref>{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}</ref>, this may be detrimental to the content coverage on “female” topics<ref>{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}</ref>, notably for social science in which women are more present. Our project proposes to improve expert contribution by making wikimedia projects (notably wikidata) useful tools that can facilitate research work, in addition to a key knowledge dissemination platform that is not country or institution-dependent. We propose to approach Wikimedia projects as a powerful (and free) knowledge management infrastructure that researchers could use. The Wikimedia ecosystem offers solutions that have strong potential to put open science principles into practices, including [[wikipedia:FAIR_data|FAIR]] principles and [[wikipedia:Linked_data#Linked_open_data|linked open data]]. == Toward a living review on just sustainability transition == === Just sustainability transition === Just sustainability transition transition is "a fair and equitable process of moving towards a post-carbon society"<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. Developping living reviews seem particularly relevant for the just transition literature: first, modeling knowledge and building graphs allows to take into account the complexity of sustainability transitions which involve multiple levels of analysis<ref name=":15" /><ref name=":16" /><ref name=":17" /> and fragmented results coming from various disciplines<ref name=":20">{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. Then, making literature reviews "living" would allow researchers to be less subject to information overload through a more systematic accumulation of knowledge. Finally, conducting this review with an open science philosophy aswers the challenge of knowledge dissemination, which is crucial in a context of socio-ecological emergency when decision-makers need to rapidely access reliable information on possible sustainability transition trajectories. === Living reviews === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1" /><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. Literature review methods are currently evolving with new technological possibilities. Generative artificial intelligence such as ChatGPT are expected to have a strong influence on literature review activities<ref name=":12">{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref name=":12" />, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but not yet integrated into tested and validated methodologies<ref name=":13">{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. Human validation stays notably necessary<ref>{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}</ref>,<ref name=":13" />. While AI can appear as a solution for scaling literature reviews, we are in the present project exploring another possible scenario which is to use more crowdsourcing in the literature review process. === Wikimedia projects === Wikipedia is a successfull example of large-scaled crowdsourcing of reliable knowledge synthesis. That is why this project proposes to explore the potential of the Wikimedia ecosystem for conducting living reviews. Since Wikipedia does aim to host original research<ref>{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&oldid=1360514388|journal=Wikipedia|language=en}}</ref>, we are working on two sister projects : Wikidata and Wikiversity. As its name suggests (wiki and data), [[wikipedia:Wikidata|Wikidata]] is, like Wikipedia, a collaborative project based on wiki editing technology; however, unlike the online encyclopaedia, it contains data rather than articles. Whilst there is a separate Wikipedia for each language version, there is only one Wikidata, containing items organised using unique identifiers. What matters when organising information in Wikidata is not the words used to define each concept, but the concept itself. Wikidata is a [[Databases|database]], more precisely a knowledge base, and a "collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation".<ref>{{Cite news|last=Chalabi|first=Mona|date=April 26, 2013|title=Welcome to Wikidata! Now what?|url=https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|access-date=October 2, 2021|archive-date=2 October 2021|archive-url=https://web.archive.org/web/20211002152920/https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|url-status=live}}</ref><ref>{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&oldid=1360462340|journal=Wikipedia|language=en}}</ref> "A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> Such graphs have a strong potential to conduct knowledge synthesis<ref name=":11" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref><ref name=":18" />. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge<ref name=":14">{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}</ref>. In complement to using Wikidata to model knowledge, we decided to use Wikiversity to report and write our research results. [[wikipedia:Wikiversity|Wikiversity]] is another Wikimedia project hosting pedagogical content, original research, and even a publishing house ([[WikiJournal|WikiJournals]])<ref>{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&oldid=1358552930|journal=Wikipedia|language=en}}</ref>. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. Our research question is : '''How can Wikimedia projects contribute to building a collaborative living review on just sustainability transition ?''' In this project, we aim to test 4 hypothesis : ● '''Hypothesis 1:''' Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations. ● '''Hypothesis 2:''' Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (e.g. conceptual typologies, cause-effect chains…). ● '''Hypothesis 3:''' SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs. ● '''Hypothesis 4''': Wikimedia or Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links (following the ideal of linked open data). We also have 2 assumptions : ● '''Assumption 1:''' Wikimedia projects have to be integrated into validated scientific protocols in order to be a valuable research tool. ● '''Assumption 2:''' Wikimedia project contribution has to be made interoperable with tools, methods and data types already used by researchers. == Methodology == Our study rely on a meta-review, that is a review of existing literature reviews. Data presented in literature reviews are usually presented as tables or diagrams, and sometimes provided as supplementary materials in publications. However, these data are not made interoperable and are not used to update prior literature reviews. Our goal was to synthesize results of previous literature reviews by making their findings compatible with linked open data and open science standards using Wikidata, Wikiversity, and other open-science infrastructures. The first step was to build and enrich the bibliographic metadata of a corpus of articles we selected into Wikidata. The second step was to model the content of these articles in Wikidata (e.g. typologies, causes-effects relationships...). The third step was to experiment relevant visualization of this content (e.g. causes-effects graphs). The las step was to write our report on a Wikiversity page, including links to our knowledge graph, following a linked open data philosophy. == 1. Building an academic corpus and enriching bibliographic metadata == The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations). To do so we imported academic references into Wikidata, and explored the advantages of constituting a scholarly corpus on Wikidata in comparison (or in complementarity) to existing tools used by researchers such as reference management softwares and knowledge management softwares. Reference management software (Zenodo, Mendeley…) are used to collect scientific item metadata and integrate them into academic writing. They can also be used to analyze and annotate academic articles and can include export functions making the data interoperable with other analysis tools. Knowledge management software (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…) are used by some researchers to organize their ideas. To build and enrich our academic corpus on Wikidata, we searched existing databases, selected the sample of articles we wanted to study, imported these articles metadata into Wikidata, enriched these metadata and finally reflected on the advantages and limitations of Wikidata to build a rich academic corpus. === Database search === Doing a systematic review on all aspects of just transition would have resulted in too many articles to review. We thus decided to first explore one aspect of justice : procedural justice. Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref> (e.g. the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />). Procedural justice can include issues of community and citizen participation in decision making, their political representation, their consultation, or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities)<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For our search, we selected keywords related to procedural justice (procedural justice OR procedural fairness OR democracy OR participation OR participatory) and keywords related to sustainability transition (sustainability OR energy OR climate) AND (transition OR transitions). We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article selection === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were : * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies/initiatives/cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors * Including justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper The files with the lists of included and excluded articles are available on the archive plateform Zenodo : https://zenodo.org/records/20749974 === Importing selected articles into Wikidata === Before importing the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in Wikidata. Next, we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through metadata enrichement === Metadatas are data describing other data. The metadata of academic items usually include title, author, publication outlet, publication date, pages, DOI, URL... and can be structured following specific standards (e.g. [[wikipedia:Dublin_Core|Dublin Core]]). In academic databases such as WOS or OpenAlex, the only metadata available regarding the content of an academic article are the abstract and sometimes keywords. However, researchers conducting literature reviews need more precise informations. An important part of literature review work can thus be about describing what the articles are about. For example, describing industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt) (e.g. <ref name=":5" />). By metadata enrichment, we mean completing metadata to include additional information about the content of an academic piece. In Wikidata, each type of information is added using a specific property. A property is the edge that links two entities in the Wikidata knowledge graph. We selected three Wikidata properties to describe the content of our selected articles : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe the geographical scope of the study. We also worked on adding {{Wikidata entity link|P50}}. ==== Adding {{Wikidata entity link|P921}} ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |}Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Adding {{Wikidata entity link|P8363}} ==== The sample of article we selected included only litterature reviews, but we wanted to describ what kind of literature review it was. We first read abstracts to identify all the different types of litterature reviews present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved some of these method items using the methodological references cited in the reviewed papers. For example, we added a statement saying that {{Wikidata entity link|Q101116078}} can have {{Wikidata entity link|Q653137}} as {{Wikidata entity link|P13391}}<ref>{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information & Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}</ref>. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |theory-driven evaluation used in evaluating social programmes |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |study of the statistical properties of combinations of studies from a meta-analytic dataset |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}For each article, we added the {{Wikidata entity link|P8363}} based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Adding {{Wikidata entity link|P6153}} ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Adding {{Wikidata entity link|P50}} ==== When scholarly metadata are imported into Wikidata, the name of authors are stored as a chain of characters and linked to the property {{Wikidata entity link|P2093}}. The property {{Wikidata entity link|P50}} allows to make a link with a Wikidata item representing the author. This avoids the problem of homonym authors by linking a unique identifyer to authors in Wikidata and linking this identifier to existing ones such as ORCID. We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers by categorizing scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages. ==== Adding {{Wikidata entity link|P6977}} ==== We explored the possibility to model review networks, that is linking review paper to reviewed papers (and not broadly cited papers). We selected one of the paper which had the most detailed list of reviewed papers ({{Wikidata entity link|Q114306483}}), then we added this list in Wikidata using Open refine. Then we snowballed this process for the reviewed papers which were also litterature reviews. === Advantages and limitations of Wikidata to build a rich living academic corpus === To share the result of our work, we exported the dataset we build on Wikidata and shared it on the open archive Zenodo : https://doi.org/10.5281/zenodo.20749973. The data is also available directly in Wikidata. The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations)'''.''' The sections below discuss the advantages and limitations of wikidata regarding this hypothesis. ==== Advantages of Wikidata ==== Key advantages of Wikidata are its flexible and collaborative nature as well as its interoperability. Wikidata ontology (that is how the data are structured) is collaboratively defined and properties can be added if relevant (after validation by the community). Compared to global databases like WOS or OpenAlex, Wikidata allows anyone to enter more metadata about each academic articles. Another notable advantage is that Wikidata items can be used as an interoperable [[wikipedia:Controlled_vocabulary|controlled vocabulary]]. For example, when we stated that the {{Wikidata entity link|P921}} of the article {{Wikidata entity link|Q114306483}} was {{Wikidata entity link|Q795757}}, "energy transition" was not just a word but a concept with its unique identifyer, linked to identifiers in other databases such as the Google Knowledge Graph ID or BNCF Thesaurus ID. Wikidata's collaborative nature is here adain an advantage. Contrary to institutional thesaurus, Wikidata allows anyone to add new concepts. This is particularly interesting as existing controlled vocabularies rarely reflect the degree of precision that researchers need in their work. The multilingual nature of Wikidata was also a strengh, some Wikidata contributors added labels for the concepts we used into different languages (For example, contributors added labels for {{Wikidata entity link|Q14944319}} in Armenian and Slovenian, languages we do not speak at all). ==== Limitations of Wikidata ==== The limitations of Wikidata is that it is not yet well integrated with the tools researchers use to do literature reviews (kowledge management softwares and bibliographic databases). Compared to reference management softwares (Zenodo, Mendeley…) and knowledge management softwares (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…), Wikidata is too general and does not allow to work on full texts. References and knowledge management softwares allow researcher to build their own specialised knowledge base, by taking notes and highlighting the content of the full texts. Wikidata is not connected to this process and there is a missing tool to facilitate the construction of graphs from the qualitative analysis of texts. In addition, when one is working on a specific corpus of item in Wikidata, it is also difficult to keep track of this corpus. We linked each academic item we were working on to our research project by adding a statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}, but it was still relatively difficult to "filter" the part of the knowledge graph we were working on. Compared to bilbiographic catalogues (OpenAlex, Web Of Science, GoTriple...), Wikidata will never be as exhaustive and do not offer user-friendly search functions. Since 2014, an important amount of bibliographic data was imported in Wikidata with the project [[d:Wikidata:WikiCite|Wikicite]]. At the time of its creation, Wikicite was adressing the issue of closed bibliographic data and was trying to make these data open, many academic items were imported automatically in Wikidata through scraping. This practice was abandoned because the large amont of bibliographic data congested queries on Wikidata (this led to the decision to split the Wikidata graph between academic and non academic entities), and because new open science initiatives, notably OpenAlex (2022), are now taking on the task of creating a exhaustive catalogues of all scholarly production. In this context, the community has to rethink the purpose of Wikidata regarding bilbiographic data. ==== Future possbilities ==== A solution to the limitations would be to developp the role of Wikidata as a link between other tools of the open science ecosystem. For example, developping and maintaining plugins or extensions for specialised softwares like Zotero, Wikibase, and Omeka could connect Wikidata with more specialised graphs. Such extensions could help building local graphs by allowing the reuse of wikidata item (eg. autocompletion), but also help contributing to Wikidata thanks to export features. (Example :<ref>{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}</ref>) Wikidata could then be an intermediary between locally curated corpus and more exhaustive bilbiographic catalogues such as OpenAlex. For example, Wikidata items could be used to tag articles in a more precise way instead of using keywords and sharing enriched corpus in Wikidata could help to train more precise taging algorythms. == 2.Modelling the content of litterature reviews == The goal of this step was to test '''Hypothesis 2''' (Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference) by modelling the content of our selected articles into Wikidata. [[wikipedia:Knowledge_modeling|Knowledge modelling]] is the process of making a machine readable model of knowledge. As we have a background in social sciences, we felt the need to question the relationship between this process and other methodologies such as concept mapping, thematic networks and causal networks. We present these methodologies before describing our current knowledge modelling experimentations. === Concept mapping, thematic networks and causal networks === This section presents social science methodology that presents similarities with knowledge modelling. ==== Concept maps ==== [[File:Conceptual_Diagram_-_Example.svg|link=https://en.wikipedia.org/wiki/File:Conceptual_Diagram_-_Example.svg|thumb|Example conceptual diagram|251x251px]]Concept maps are ''concepts'' (boxes) and ''propositions'' (arrow indicating the relationship between two boxes)<ref name=":19">Cañas, Alberto J., et al. "CmapTools: A knowledge modeling and sharing environment." (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf</ref>. Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]<ref name=":19" />, Altas.Ti "network" feature...). The "box and arrow" logic is similar to how knowledge is modelled on Wikidata : the equivalent of concepts is ''item'' and the equivalent of propositions are ''statements''. The difference between a softwares like Cmap and Wikidata is the underlying format of the data. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|447x447px|Structure of a thematic network (Source: based on Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. [[File:Adoption_CLD.svg|link=https://en.wikipedia.org/wiki/File:Adoption_CLD.svg|thumb|421x421px|Causal loop diagram of ''Adoption'' model, used to demonstrate systems dynamics]] ==== Causal diagrams ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref name=":21">{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> After identifying themes, qualitative researchers are expected to theorize the "arrows" between themes<ref name=":21" />R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. === Knowledge modelling in Wikidata === This section presents our knowledge modelling experimentation in Wikidata. ==== Conceptual modelling ==== Our first step was to reflect on what is a "concept" and what kind of wikidata properties could be used to model concepts in Wikidata. Scholars in management have called for more rigorous ways to define concepts<ref name=":22" /> and modelling concepts in Wikidata could help to build less ambiguous concepts. Concept definition encompass various aspects such as the nature of the phenomenon, its characteristics, the links with prototypical cases or examples, the contrast with other concepts, the links with causes and consequences...<ref name=":22">{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}</ref>, and scholars have advised to take insight from philosophy to work on concepts<ref>{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}</ref>. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref>. We summarize these approaches below and examine which wikidata properties exist to represent them. *Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... *Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. (The closer a phenomenon is to the prototype, the more likely it belong to the category). Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}} (see discussion here https://www.wikidata.org/wiki/Help:Basic_membership_properties). *Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}, {{P|1478}}, {{P|P9353}} (see discussions here : https://www.wikidata.org/wiki/Help:Modeling_causes/en). *Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can use it before really knowing what its properties will be. *Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Testing concept modelling on {{Wikidata entity link|Q14944319}} ==== To test concept modelling, we started by experimenting with the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic : *{{Wikidata entity link|Q137901202}} *{{Wikidata entity link|Q137901196}} *{{Wikidata entity link|Q137901182}} *{{Wikidata entity link|Q136447761}} *{{Wikidata entity link|Q129652515}} *{{Wikidata entity link|Q114306483}} We read each paper and used them as source in manually entered statements in the item {{Wikidata entity link|Q14944319}}. For example, Droubi et. Al stated "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}} and {{Wikidata entity link|Q3249551}}, using the paper as reference (see screenshot below). The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&oldid=2495982191. [[File:Wikidata statement- energy democracy is an instance of ideal.png|915x915px|border]] We listed the difficulties encountered as we worked and we also asked the Wikidata community to give us feedback on our modelling on the item discussion page (https://www.wikidata.org/wiki/Talk:Q14944319). These issues were related to contradictions, precision, concision and ontology. We discuss each issue and draft recommandations to refine our modelling process. ===== Contradictions ===== Wikidata contributor's feedback highlighted some apparent contradictions (The values in "does not have effect" seems contrary to what is listed in "has goal".) We would however argue this is not a problem because "statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata"<ref>{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}</ref>. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref>.Besides, in the case of goals versus effects statements, it is not contradictory because one can have a goal and fail to achieve it. In the case of energy democracy, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing<ref name=":20" />. Recommandations : Contradictions are allowed in Wikidata. ===== Precision ===== We noted that conceptual modelling requires an important degree of formalization and precision. This is a key advantage of Wikidata to be able to create links toward precise concepts which have their own identifiers. For example, we were able to create statements about specific laws and their unique identifiers in legal databases (e.g. {{Wikidata entity link|Q139764294}} and its identifier in the EUR-Lex database). However, the sources we are working with are not always precise enough and when concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, {{Wikidata entity link|Q12888920}}... We can see here that academic texts are using natural language and thus are using ambiguous terms. As a result, we received feedbacks regarding a lack of precision in our statements (too many and too vague statements). In addition to the ambiguity of sources, a reason why we ended up with very general statements is because we avoided the creation of new Wikidata items. While following this implicit rule allowed us to focus on the most notable concepts, creating new items could also help make the statements more precise. Recommandations : If a concept is ambiguous it should not be included in the modelled statements. Create more precise relevant concepts if they do not exist in Wikidata. [[File:Wikidata visualisation screenshot of subclasses relationships including the item political concept.png|thumb|298x298px|Subclass relationships between "concept" and "political concept".]] ===== Concision ===== Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were "in the tree of another value". The rule we take from this feeback is a need of logical simplification. Two examples illustrate possible logical simplification : *We stated that {{Wikidata entity link|Q14944319}} was an {{Wikidata entity link|P31}} {{Wikidata entity link|Q33104069}} and an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}. But in that case, it is not necessary to state that it is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}, because {{Wikidata entity link|Q33104069}} is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q131362181}}, which is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q151885}} (see diagram on the right). Here, we have to keep only the more precise item. This reasonning is based on the assumption that {{Wikidata entity link|P279}} is transitive. It seems this reasonning could be generalized (we opened a discussion about this here https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning/Use_cases#Parcimonious_statement_constraints_based_on_subclass_of_(P279)_and_part_of_(P361)_transitivity<nowiki/>) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using "instance of" as transversal property and checking "instance of or subclass of"). *We stated that {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P2670}} {{Wikidata entity link|Q15991216}} and {{Wikidata entity link|Q113514984}}. But if we consider that {{Wikidata entity link|Q15991216}} is a {{Wikidata entity link|P279}} of {{Wikidata entity link|Q113514984}}, then the inclusion of {{Wikidata entity link|Q15991216}} is implied. Here we could keep only the broader item {{Wikidata entity link|Q113514984}}. But unlike the reasonning above, we could not identify a way to generalize this reasonning. For example, {{Wikidata entity link|Q113514984}} can be a subclass of {{Wikidata entity link|Q43229}}, but using the former item is more informative than using the later. For the property {{Wikidata entity link|P2670}}, using a value that is too precise results in too many statements and using a value that is too general results in too trivial statements. Choosing the degree of precision is a problem similar to choosing whether or not to create a new item. Recommandations : When possible, use logical simplification to make statements more parcimonious. The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize redundancies. ===== Quantification ===== Modelling quantitative statements was challenging. We mostly skipped those but made an attempt for one case : the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources"<ref>{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research & Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}</ref>. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for "final energy consumption". There are such properties in Wikidata : for example, for renewable energy subsidies, there is {{Wikidata entity link|P6826}}. This type of property is notably useful to display information about an item (e.g. the population of a town) on Wikipedia through dynamic infoboxes that are updated with Wikidata information. But the current informal rule on Wikidata property creation is to prefer the use of general properties and avoid the creation of specific properties. Therefore creating properties for every quantifiable characteristic may not be possible. An alternative way to model quantity is to create a specific item to represent what is quantified and use other properties to represent quantities (see example below). [[File:Wikidata screenshot of percentage modelling.png|border|783x783px]] However, Wikidata is optimized for statements with a few discrete values and it is more convenient to store quantitative (tabular) data in elsewhere<ref>{{Cite journal|last=Larrañaga|first=Galder Gonzalez|last2=Pintscher|first2=Lydia|last3=Ainali|first3=Jan|last4=Fauconnier|first4=Sandra|date=July 2026|title=Tabular data vs structured data.pptx|url=https://docs.google.com/presentation/d/1k37UX-VPwMgJbXoB7ZxsztE5LdKR_isM|journal=Wikimania 2026}}</ref>. Instead, it might be more useful to link the item that is quantified to an external dataset about it. This can be done using the property {{Wikidata entity link|P1325}}. Recommandation : Use existing properties for the quantity if it exists. If it does not exist, create an item representing what is measured, link it toward an external datasets and add quantitative statements if relevant. ===== Ontology ===== Ultimately, several of our difficulties were linked to some ontology challenges reflecting the complexity of the concept we were working on. According to the litterature, {{Wikidata entity link|Q14944319}} refer to multiple types of entities. It represents a concept, an ideal, a process and an outcome. We reflected this with multiple {{Wikidata entity link|P31}} statements, but encountered struggles to model differences between ''process versus outcomes'', ''ideal versus reality'', ''phenomenon versus theory'' and ''discourses versus practices''. The wikidata community recommands having different items for ''process and outcome'' (criterion "{{Wikidata entity link|Q127270577}}"). For example, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork). However, this distinction is less straightforward for social processes which are are ongoing and evolving without a clear ending. Another problem was to separate ''ideal versus reality,'' we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes attempting to reach it. It was also difficult to separate ''phenomenon versus theory,'' for example, we did not manage to model the idea that the literature on energy democracy is fragmented. Finally, we also would have needed more distinctions between ''discourses and practices.'' Our sources suggested that energy democracy discourses and practices may have different causes, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movments promoting it, but this was not a very precise way to model this idea. Our first attempt to multiple "natures" of energy democracy into a single item is not satisfactory, and implementing the modelling recommandations we identified above may not be sufficient to build more parcimonious statements. Creating new items to reflect the different aspects of energy democracy may here be necessary. To do so we have to decide how to split the different natures {{Wikidata entity link|Q14944319}} and what will be the relationship between the resulting items. Here the definition of an {{Wikidata entity link|Q324254}} (formal representation) in Wikidata can quickly escalate into questionning {{Wikidata entity link|Q44325}} (metaphysical reflexion on the nature of things), and especially {{Wikidata entity link|Q1713511}} (the nature of the social world). Indeed, the relations between the different types of entities constituting the social worls are not simple nor consensual among philosophers of social science. For example, {{Wikidata entity link|Q15080858}} posits that different things have different ways of being (modes of reality). They propose to classify entities in four categories : material entities (that can exist independently of humans), conceptual entities (concepts, discourses, ideas, meaning…), artefactual entities (human-made and combining conceptual and material elements) and social entities (that depends on human activity to exist)<ref>Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.</ref>. A complex concept like {{Wikidata entity link|Q14944319}} is concerned with all these types of entities. The energy system include many material entities such as oil fields, the sun, seas, trees... and artefacts such as energy production unit, power lines, home appliances, trucks... (Including conceptual entities such as the name of these artefacts or the knowledge to make them function.). There are the social entities in which they are encompassed (the enregy sectors, energy businesses, energy policies...). There are conceptual entities like normative/political discourses discussing how these artefact and social system should work and there are conceptual entities in the academic sphere building theories about how all this works or should work. In theory, such ontology can be implemented in Wikidata as it "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8" /> Current Wikidata ontology is structured with dischotomies such as {{Wikidata entity link|Q7048977}}/{{Wikidata entity link|Q4406616}}, {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}, {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q23958946}}, {{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}, {{Wikidata entity link|Q30241068}}/{{Wikidata entity link|Q3799040}}... A quick search seem to show that entities related to the social reality can quickly fall into heterogeneous and sometime contradictory root classes (see table below). {| class="wikitable" |+ !Item !Existing root class entity |- |{{Wikidata entity link|Q190539}} |{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q67518978}} |- |{{Wikidata entity link|Q34770}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q9081}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q43229}} |{{Wikidata entity link|Q99527517}}{{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q103940464}} |- |{{Wikidata entity link|Q49773}} |{{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}} {{Wikidata entity link|Q99527517}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q12705}} |{{Wikidata entity link|Q4406616}}/{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}} {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q132907471}} |} === Future steps === Current {{Wikidata entity link|Q3882785}} in Wikidata do not seem very informative to classify the aspects of our concept in a top-down manner. Our next steps will thus be to : * Export the existing model of the concept to keep track of each modelling steps (files and graphs screenshots) *Explore if existing standards offer more relevant classes to categorize social entities (we opened the discussion about social ontology with the wikidata community here https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Ontology#Social_ontology). **CRMSoc : https://cidoc-crm.org/crmsoc/fm_releases **Criteria for choosing an ontology = choose something that is used and maintained by a community (colleague feedback) *Create new items to reflect the different facets of {{Wikidata entity link|Q14944319}} : the social movement advocating for it, the political concept theorizing it, the concrete initiatives implementing it, the litterature theorizing it, etc... based on a new reading of our sample of article and aligned with existing ontologies. *Compare grounded theory/CAQDAS with knowledge modelling methods, test annotation softwares : https://inception-project.github.io/ ; https://pacte.crim.ca/index_en.html *Read articles again and model more precise statements, including quotation of the original text in the reference field. *Write a more precise research question to orient modelling : What are the main concepts of just transition, their definition and their relationships ? * == 3. Data visualisation == The goal of this step is to test '''Hypothesis 3''' (SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs). At this stage we started to identify visualization use cases we would like to test, they are listed below. === Filter statements === *Visualize only statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH *Visualize only items which are part of the present project (all the academic items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Properties visualisation === *Visualise the "tree" of a property used in an item : Wikidata graph builder seem to be the most user friendly, robust and versatile tool to visualise a graph of a single property (https://angryloki.github.io) === Concepts visualisation === *Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request "topic in context") *Map the statements that two items have in common to make comparisons. === Mapping sources consensus === *Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible. == 4. Writing == The goal of this step is to test '''Hypothesis 4''' (Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links). * To do idea : write a Wikiversity page for each central concept (e.g. energy democracy, energy community, energy citizenship...) === Advantages of Wikiversity === Writing on a Wikiversity page offers some advantages to implement the principles of open linked data in text format. We could cite academic items using their Wikidata QID to generate the citations below (but if we use an URL to cite a paper, it does not automatically check if there is an existing QID for it), and also link toward Wikidata entities using a template ([[Template:Wikidata entity link|Wikidata entity link]]). Although we did not translate the present page for now, Wikiversity offers the possibility to translate a page in multiple languages (an interesting possibility in the context of the The FAIRisation of scholarly communication<ref>{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}</ref>). The possibility to view the page history provide an exhaustive versionning of a paper, which constitute interesting data documenting the academic writing process (at first we did not comment the nature of our modifications but after thinking about this reuse possibility, we described it with more details). The contribution statistics based on the history also offers a new way to track author's contributions to a paper in a transparent way. Some researchers experimented relationships between the wikimedia community and researchers by making researchers review wikipedia articles<ref>{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}</ref>. === The issue of text interoperability === A key issue we are encountering is the question of the interoperability of texts. While the interoperability of data is starting to be well discussed in the open science community, the interoperability of texts do not seem to benefit from the same level of discussion. We encountered several interoperability issues regarding our writing. First, copying texts written on a word processor software (e.g. microsoft word) into a wiki page (or the other way around) is relatively seamless in terms of formatting, but the management of references is lacking. Reformatting references is very time consuming and a real barrier for text interoperability in academic context : it is difficult to copy text from an academic publication into a wiki text, and difficult to turn a wiki text into a publication (the only possibility is to export the page as a PDF and you may be reading this article in this format). There are also uncertaineties regarding how to reuse texts published under creative common licences. Academic texts published under CC-BY-SA licences can in theory be remixed and reused. But academia does not have established practices regarding how this can be done. If we want to reuse a whole page, should we put it in quotation marks and simply cite the paper ? Should the original authors be listed as co-authors ? To what extent academic publisher will accept to publish "remixed" texts while they usually require that publications contain mainly unpublished content ? The norms of what is appropriate remix and reuse practices in academia has yet to be decided... and this could be an interesting conversation to have in the open science community. == Discussion == Our goal was to assess the potential of Wikidata and Wikiversity to build a living literature review method and tackle issues of information overload, knowledge synthesis and knowledge dissemination, following open science principles. By conducting a meta-review on just sustainability transition (that is a review of existing literature reviews), and experimenting with existing technical solutions, we were able to identify the strenghs and limitations of the Wikidata ecosystem. First Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations (Hypothesis 1 is verified). However, the technical barriers to do so are still high and the Wikidata database is too general to allow a community to work on a specific curated corpus of scientific items. Secondly, Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (Hypothesis 2 is verified). However, this requires developping new methodological standards regarding what is a rigorous modelling in social science, as well as building specific {{Wikidata entity link|Q324254}}. An unexpected finding is that the formalization necessary for knowledge modelling in wikidata invites to formalize the reasonning behind modelling choices, a process that is rarely done in qualitative analysis even though such reasonning are likely present. The modelling process also responds to calls in management sciences to engage in more systematic categorizing to avoid semantic confusion<ref>{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}</ref>. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to "navigate" it (Hypothesis 3 is partially verified). SPARQL-based queries are powerful but they require technical knowledge, especially now that the split between academic and non-academic items in Wikidata requires to write federated queries. To be able to "navigate" scientific corpora and scientific knowledge graphs on Wikidata, more user-friendly tools would be needed. Finally, we did see advantaged in using Wikiversity pages to write collaboratively in text format augmented by hypertext links, but there are still important technical and instittional barriers (Hypothesis 3 is partially verified) : the interoperability of text is a key issue to reuse publications. The main issues we encountered were the management of references and uncertainty regarding the compatibilities between writing publicly on a wiki page and engaging in classical publication processes. A potential solution to the issues encountered could be to develop a specialised literature review software that would allow researchers to build living literature reviews including knowledge graphs and wiki pages. This kind of tool should be user-friendly and include the missing tools we identified. In addition, it could include more advanced analysis functions such as logical reasonning based on the knowledge graph<ref name=":9" />. === Future research === - Can we "reason" with Wikidata (See: https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning) : neurosymbolic AI ? == Funding == This project is funded by the [[m:Grants:Programs/Wikimedia_Research_&_Technology_Fund/Wikimedia_Research_Fund|Wikimedia Research Fund]], Grant ID: G-RS-2504-18935. The text of the initial research proposal is available here : https://doi.org/10.5281/zenodo.20760603. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 0wd4vk6um9ag2vrwwuasdiwxxc1clli 2819669 2819648 2026-07-28T15:11:21Z Jeanne Noiraud 1366702 /* Future steps */ writing decision to leave out upper ontology discussions + writing next steps 2819669 wikitext text/x-wiki == Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) == == Acknowledgements == The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. === Contributors === {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling, article writing |- |Amélie E. Pereira |Laboratoire DICEN IDF |https://orcid.org/0009-0005-5928-5586 |Meta-data enrichement, article writing |- |Finn Nielsen |Technical University of Denmark |https://orcid.org/0000-0001-6128-3356 |Data visualisation |} Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review == Introduction == Just sustainability transition refers to the process of shifting towards sustainable practices in a way that is equitable and inclusive. It includes dimensions of procedural, recognition, distributive and reparative justice and the concept is related to climate justice, environmental justice and energy justice<ref>{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}</ref><ref>{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}</ref>. The study of sustainability transitions in social sciences requires dynamic and adaptive research synthesis methods. Sustainability transitions involve complex, multi-level processes influenced by technological, economic, social, and policy factors<ref name=":15">{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}</ref><ref name=":16">{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}</ref><ref name=":17">{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research & Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}</ref>. Given the rapidly evolving nature of sustainability-related research, static literature reviews often become outdated, limiting their usefulness for policymakers, scholars, and practitioners. A living literature review – continuously updated with new findings – ensures that emerging insights, case studies, and theoretical developments are integrated cumulatively into the knowledge base. Developing such review will answer the call for more evidence-based practices in management sciences<ref>{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning & Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}</ref><ref>Pfeffer, J., & Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. </ref>. Our project assesses the potential of Wikidata to build living review workflow on sustainability transition. We address three issues encountered by scientists: information overload, knowledge synthesis and results dissemination. === The problem of academic information overload === Global scientific output is quickly growing<ref>{{Cite journal|last=Bornmann|first=Lutz|last2=Mutz|first2=Rüdiger|date=2015|title=Growth rates of modern science: A bibliometric analysis based on the number of publications and cited references|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/asi.23329|journal=Journal of the Association for Information Science and Technology|language=en|volume=66|issue=11|pages=2215–2222|doi=10.1002/asi.23329|issn=2330-1643}}</ref><ref name=":23">{{Cite journal|last=Hanson|first=Mark A.|last2=Barreiro|first2=Pablo Gómez|last3=Crosetto|first3=Paolo|last4=Brockington|first4=Dan|date=2024-11-01|title=The strain on scientific publishing|url=https://doi.org/10.1162/qss_a_00327|journal=Quantitative Science Studies|volume=5|issue=4|pages=823–843|doi=10.1162/qss_a_00327|issn=2641-3337}}</ref><ref>{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}</ref>, pushed by the “publish or perish” model incentivizing researchers to increase the quantity of research outputs. Researchers are subject to information overload as the number of publications to read is beyond what a human brain can handle, they are expected to produce high-quality research under an increasing time pressure<ref name=":23" />. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge<ref>{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}</ref>. “Wikifying science” may in this context contribute to facilitating researcher’s work while preserving scientific quality. That is why in this project, we aim to build a searchable academic publication database with enriched meta-data that will allow scholars to navigate the existing publications corpus related to just sustainability transition more easily. === The problem of knowledge synthesis === The volume of academic production is rendering knowledge synthesis difficult. Scholars have thus called for making literature reviews cumulative and updatable<ref>{{Citation|title=Day 2 {{!}} Arnaud Vaganay: Reproducible Literature Reviews|url=https://www.youtube.com/watch?v=Nspd_1cx9kc|date=2017-10-19|accessdate=2026-06-23|last=Berkeley Initiative for Transparency in the Social Sciences (BITSS)}}</ref><ref>{{Cite journal|last=L.|first=Mobley, David|last2=M.|first2=Zuckerman, Daniel|date=2015-02-03|title=A proposal for regularly updated review/survey articles: "Perpetual Reviews"|url=https://arxiv.org/abs/1502.01329|journal=arXiv.org|language=en|doi=10.48550/arXiv.1502.01329}}</ref> and for shifting from static text format publications to dynamic knowledge mapping<ref name=":11">{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}</ref>. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : <ref>{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}</ref>,<ref>{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}</ref>,<ref name=":18">{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}</ref>). While such reviews method exist for quantitative research producing standardized results, they are not adapted to synthetize social science studies on sustainability transitions that involve diverse methodologies and various disciplinary perspectives. The goal of the project is to propose a demonstration of a living review method for social science findings on just sustainability transition, relying on the collaborative model and tools of Wikimedia projects notably Wikidata, Wikiversity and Wikipedia. === The problem of scientific results dissemination === There is urgent need to disseminate knowledge on impactful topics like sustainability transition while proprietary publication models, disinformation and censorship (e.g. US) is threatening access to free and reliable knowledge. In parallel, social scientists struggle to make their work impactful<ref>{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}</ref>. Wikipedia is a key knowledge dissemination platform widely used by students<ref>{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}</ref> and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature<ref>{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}</ref> and that some scholars cite directly Wikipedia<ref>{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}</ref>. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives<ref>{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}</ref>,<ref>{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}</ref>, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow<ref name=":10">Taraborelli, D., Mietchen, D., Alevizou, P., & Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. <nowiki>http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf</nowiki> </ref>. In addition, expert participation is not immune to the gender gap<ref name=":10" />. Because of gender segregation in disciplines<ref>{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}</ref>, this may be detrimental to the content coverage on “female” topics<ref>{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}</ref>, notably for social science in which women are more present. Our project proposes to improve expert contribution by making wikimedia projects (notably wikidata) useful tools that can facilitate research work, in addition to a key knowledge dissemination platform that is not country or institution-dependent. We propose to approach Wikimedia projects as a powerful (and free) knowledge management infrastructure that researchers could use. The Wikimedia ecosystem offers solutions that have strong potential to put open science principles into practices, including [[wikipedia:FAIR_data|FAIR]] principles and [[wikipedia:Linked_data#Linked_open_data|linked open data]]. == Toward a living review on just sustainability transition == === Just sustainability transition === Just sustainability transition transition is "a fair and equitable process of moving towards a post-carbon society"<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. Developping living reviews seem particularly relevant for the just transition literature: first, modeling knowledge and building graphs allows to take into account the complexity of sustainability transitions which involve multiple levels of analysis<ref name=":15" /><ref name=":16" /><ref name=":17" /> and fragmented results coming from various disciplines<ref name=":20">{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. Then, making literature reviews "living" would allow researchers to be less subject to information overload through a more systematic accumulation of knowledge. Finally, conducting this review with an open science philosophy aswers the challenge of knowledge dissemination, which is crucial in a context of socio-ecological emergency when decision-makers need to rapidely access reliable information on possible sustainability transition trajectories. === Living reviews === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1" /><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. Literature review methods are currently evolving with new technological possibilities. Generative artificial intelligence such as ChatGPT are expected to have a strong influence on literature review activities<ref name=":12">{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref name=":12" />, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but not yet integrated into tested and validated methodologies<ref name=":13">{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. Human validation stays notably necessary<ref>{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}</ref>,<ref name=":13" />. While AI can appear as a solution for scaling literature reviews, we are in the present project exploring another possible scenario which is to use more crowdsourcing in the literature review process. === Wikimedia projects === Wikipedia is a successfull example of large-scaled crowdsourcing of reliable knowledge synthesis. That is why this project proposes to explore the potential of the Wikimedia ecosystem for conducting living reviews. Since Wikipedia does aim to host original research<ref>{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&oldid=1360514388|journal=Wikipedia|language=en}}</ref>, we are working on two sister projects : Wikidata and Wikiversity. As its name suggests (wiki and data), [[wikipedia:Wikidata|Wikidata]] is, like Wikipedia, a collaborative project based on wiki editing technology; however, unlike the online encyclopaedia, it contains data rather than articles. Whilst there is a separate Wikipedia for each language version, there is only one Wikidata, containing items organised using unique identifiers. What matters when organising information in Wikidata is not the words used to define each concept, but the concept itself. Wikidata is a [[Databases|database]], more precisely a knowledge base, and a "collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation".<ref>{{Cite news|last=Chalabi|first=Mona|date=April 26, 2013|title=Welcome to Wikidata! Now what?|url=https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|access-date=October 2, 2021|archive-date=2 October 2021|archive-url=https://web.archive.org/web/20211002152920/https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|url-status=live}}</ref><ref>{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&oldid=1360462340|journal=Wikipedia|language=en}}</ref> "A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> Such graphs have a strong potential to conduct knowledge synthesis<ref name=":11" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref><ref name=":18" />. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge<ref name=":14">{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}</ref>. In complement to using Wikidata to model knowledge, we decided to use Wikiversity to report and write our research results. [[wikipedia:Wikiversity|Wikiversity]] is another Wikimedia project hosting pedagogical content, original research, and even a publishing house ([[WikiJournal|WikiJournals]])<ref>{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&oldid=1358552930|journal=Wikipedia|language=en}}</ref>. Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and a history log tab. Our research question is : '''How can Wikimedia projects contribute to building a collaborative living review on just sustainability transition ?''' In this project, we aim to test 4 hypothesis : ● '''Hypothesis 1:''' Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations. ● '''Hypothesis 2:''' Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (e.g. conceptual typologies, cause-effect chains…). ● '''Hypothesis 3:''' SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs. ● '''Hypothesis 4''': Wikimedia or Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links (following the ideal of linked open data). We also have 2 assumptions : ● '''Assumption 1:''' Wikimedia projects have to be integrated into validated scientific protocols in order to be a valuable research tool. ● '''Assumption 2:''' Wikimedia project contribution has to be made interoperable with tools, methods and data types already used by researchers. == Methodology == Our study rely on a meta-review, that is a review of existing literature reviews. Data presented in literature reviews are usually presented as tables or diagrams, and sometimes provided as supplementary materials in publications. However, these data are not made interoperable and are not used to update prior literature reviews. Our goal was to synthesize results of previous literature reviews by making their findings compatible with linked open data and open science standards using Wikidata, Wikiversity, and other open-science infrastructures. The first step was to build and enrich the bibliographic metadata of a corpus of articles we selected into Wikidata. The second step was to model the content of these articles in Wikidata (e.g. typologies, causes-effects relationships...). The third step was to experiment relevant visualization of this content (e.g. causes-effects graphs). The las step was to write our report on a Wikiversity page, including links to our knowledge graph, following a linked open data philosophy. == 1. Building an academic corpus and enriching bibliographic metadata == The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations). To do so we imported academic references into Wikidata, and explored the advantages of constituting a scholarly corpus on Wikidata in comparison (or in complementarity) to existing tools used by researchers such as reference management softwares and knowledge management softwares. Reference management software (Zenodo, Mendeley…) are used to collect scientific item metadata and integrate them into academic writing. They can also be used to analyze and annotate academic articles and can include export functions making the data interoperable with other analysis tools. Knowledge management software (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…) are used by some researchers to organize their ideas. To build and enrich our academic corpus on Wikidata, we searched existing databases, selected the sample of articles we wanted to study, imported these articles metadata into Wikidata, enriched these metadata and finally reflected on the advantages and limitations of Wikidata to build a rich academic corpus. === Database search === Doing a systematic review on all aspects of just transition would have resulted in too many articles to review. We thus decided to first explore one aspect of justice : procedural justice. Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref> (e.g. the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />). Procedural justice can include issues of community and citizen participation in decision making, their political representation, their consultation, or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities)<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For our search, we selected keywords related to procedural justice (procedural justice OR procedural fairness OR democracy OR participation OR participatory) and keywords related to sustainability transition (sustainability OR energy OR climate) AND (transition OR transitions). We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article selection === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were : * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies/initiatives/cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors * Including justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper The files with the lists of included and excluded articles are available on the archive plateform Zenodo : https://zenodo.org/records/20749974 === Importing selected articles into Wikidata === Before importing the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in Wikidata. Next, we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through metadata enrichement === Metadatas are data describing other data. The metadata of academic items usually include title, author, publication outlet, publication date, pages, DOI, URL... and can be structured following specific standards (e.g. [[wikipedia:Dublin_Core|Dublin Core]]). In academic databases such as WOS or OpenAlex, the only metadata available regarding the content of an academic article are the abstract and sometimes keywords. However, researchers conducting literature reviews need more precise informations. An important part of literature review work can thus be about describing what the articles are about. For example, describing industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt) (e.g. <ref name=":5" />). By metadata enrichment, we mean completing metadata to include additional information about the content of an academic piece. In Wikidata, each type of information is added using a specific property. A property is the edge that links two entities in the Wikidata knowledge graph. We selected three Wikidata properties to describe the content of our selected articles : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe the geographical scope of the study. We also worked on adding {{Wikidata entity link|P50}}. ==== Adding {{Wikidata entity link|P921}} ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |}Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Adding {{Wikidata entity link|P8363}} ==== The sample of article we selected included only litterature reviews, but we wanted to describ what kind of literature review it was. We first read abstracts to identify all the different types of litterature reviews present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved some of these method items using the methodological references cited in the reviewed papers. For example, we added a statement saying that {{Wikidata entity link|Q101116078}} can have {{Wikidata entity link|Q653137}} as {{Wikidata entity link|P13391}}<ref>{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information & Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}</ref>. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |theory-driven evaluation used in evaluating social programmes |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |study of the statistical properties of combinations of studies from a meta-analytic dataset |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}For each article, we added the {{Wikidata entity link|P8363}} based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Adding {{Wikidata entity link|P6153}} ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Adding {{Wikidata entity link|P50}} ==== When scholarly metadata are imported into Wikidata, the name of authors are stored as a chain of characters and linked to the property {{Wikidata entity link|P2093}}. The property {{Wikidata entity link|P50}} allows to make a link with a Wikidata item representing the author. This avoids the problem of homonym authors by linking a unique identifyer to authors in Wikidata and linking this identifier to existing ones such as ORCID. We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers by categorizing scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages. ==== Adding {{Wikidata entity link|P6977}} ==== We explored the possibility to model review networks, that is linking review paper to reviewed papers (and not broadly cited papers). We selected one of the paper which had the most detailed list of reviewed papers ({{Wikidata entity link|Q114306483}}), then we added this list in Wikidata using Open refine. Then we snowballed this process for the reviewed papers which were also litterature reviews. === Advantages and limitations of Wikidata to build a rich living academic corpus === To share the result of our work, we exported the dataset we build on Wikidata and shared it on the open archive Zenodo : https://doi.org/10.5281/zenodo.20749973. The data is also available directly in Wikidata. The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations)'''.''' The sections below discuss the advantages and limitations of wikidata regarding this hypothesis. ==== Advantages of Wikidata ==== Key advantages of Wikidata are its flexible and collaborative nature as well as its interoperability. Wikidata ontology (that is how the data are structured) is collaboratively defined and properties can be added if relevant (after validation by the community). Compared to global databases like WOS or OpenAlex, Wikidata allows anyone to enter more metadata about each academic articles. Another notable advantage is that Wikidata items can be used as an interoperable [[wikipedia:Controlled_vocabulary|controlled vocabulary]]. For example, when we stated that the {{Wikidata entity link|P921}} of the article {{Wikidata entity link|Q114306483}} was {{Wikidata entity link|Q795757}}, "energy transition" was not just a word but a concept with its unique identifyer, linked to identifiers in other databases such as the Google Knowledge Graph ID or BNCF Thesaurus ID. Wikidata's collaborative nature is here adain an advantage. Contrary to institutional thesaurus, Wikidata allows anyone to add new concepts. This is particularly interesting as existing controlled vocabularies rarely reflect the degree of precision that researchers need in their work. The multilingual nature of Wikidata was also a strengh, some Wikidata contributors added labels for the concepts we used into different languages (For example, contributors added labels for {{Wikidata entity link|Q14944319}} in Armenian and Slovenian, languages we do not speak at all). ==== Limitations of Wikidata ==== The limitations of Wikidata is that it is not yet well integrated with the tools researchers use to do literature reviews (kowledge management softwares and bibliographic databases). Compared to reference management softwares (Zenodo, Mendeley…) and knowledge management softwares (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…), Wikidata is too general and does not allow to work on full texts. References and knowledge management softwares allow researcher to build their own specialised knowledge base, by taking notes and highlighting the content of the full texts. Wikidata is not connected to this process and there is a missing tool to facilitate the construction of graphs from the qualitative analysis of texts. In addition, when one is working on a specific corpus of item in Wikidata, it is also difficult to keep track of this corpus. We linked each academic item we were working on to our research project by adding a statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}, but it was still relatively difficult to "filter" the part of the knowledge graph we were working on. Compared to bilbiographic catalogues (OpenAlex, Web Of Science, GoTriple...), Wikidata will never be as exhaustive and do not offer user-friendly search functions. Since 2014, an important amount of bibliographic data was imported in Wikidata with the project [[d:Wikidata:WikiCite|Wikicite]]. At the time of its creation, Wikicite was adressing the issue of closed bibliographic data and was trying to make these data open, many academic items were imported automatically in Wikidata through scraping. This practice was abandoned because the large amont of bibliographic data congested queries on Wikidata (this led to the decision to split the Wikidata graph between academic and non academic entities), and because new open science initiatives, notably OpenAlex (2022), are now taking on the task of creating a exhaustive catalogues of all scholarly production. In this context, the community has to rethink the purpose of Wikidata regarding bilbiographic data. ==== Future possbilities ==== A solution to the limitations would be to developp the role of Wikidata as a link between other tools of the open science ecosystem. For example, developping and maintaining plugins or extensions for specialised softwares like Zotero, Wikibase, and Omeka could connect Wikidata with more specialised graphs. Such extensions could help building local graphs by allowing the reuse of wikidata item (eg. autocompletion), but also help contributing to Wikidata thanks to export features. (Example :<ref>{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}</ref>) Wikidata could then be an intermediary between locally curated corpus and more exhaustive bilbiographic catalogues such as OpenAlex. For example, Wikidata items could be used to tag articles in a more precise way instead of using keywords and sharing enriched corpus in Wikidata could help to train more precise taging algorythms. == 2.Modelling the content of litterature reviews == The goal of this step was to test '''Hypothesis 2''' (Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference) by modelling the content of our selected articles into Wikidata. [[wikipedia:Knowledge_modeling|Knowledge modelling]] is the process of making a machine readable model of knowledge. As we have a background in social sciences, we felt the need to question the relationship between this process and other methodologies such as concept mapping, thematic networks and causal networks. We present these methodologies before describing our current knowledge modelling experimentations. === Concept mapping, thematic networks and causal networks === This section presents social science methodology that presents similarities with knowledge modelling. ==== Concept maps ==== [[File:Conceptual_Diagram_-_Example.svg|link=https://en.wikipedia.org/wiki/File:Conceptual_Diagram_-_Example.svg|thumb|Example conceptual diagram|251x251px]]Concept maps are ''concepts'' (boxes) and ''propositions'' (arrow indicating the relationship between two boxes)<ref name=":19">Cañas, Alberto J., et al. "CmapTools: A knowledge modeling and sharing environment." (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf</ref>. Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]<ref name=":19" />, Altas.Ti "network" feature...). The "box and arrow" logic is similar to how knowledge is modelled on Wikidata : the equivalent of concepts is ''item'' and the equivalent of propositions are ''statements''. The difference between a softwares like Cmap and Wikidata is the underlying format of the data. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|447x447px|Structure of a thematic network (Source: based on Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. [[File:Adoption_CLD.svg|link=https://en.wikipedia.org/wiki/File:Adoption_CLD.svg|thumb|421x421px|Causal loop diagram of ''Adoption'' model, used to demonstrate systems dynamics]] ==== Causal diagrams ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref name=":21">{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> After identifying themes, qualitative researchers are expected to theorize the "arrows" between themes<ref name=":21" />R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. === Knowledge modelling in Wikidata : first round of analysis === This section presents our knowledge modelling experimentation in Wikidata. ==== Conceptual modelling ==== Our first step was to reflect on what is a "concept" and what kind of wikidata properties could be used to model concepts in Wikidata. Scholars in management have called for more rigorous ways to define concepts<ref name=":22" /> and modelling concepts in Wikidata could help to build less ambiguous concepts. Concept definition encompass various aspects such as the nature of the phenomenon, its characteristics, the links with prototypical cases or examples, the contrast with other concepts, the links with causes and consequences...<ref name=":22">{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}</ref>, and scholars have advised to take insight from philosophy to work on concepts<ref>{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}</ref>. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref>. We summarize these approaches below and examine which wikidata properties exist to represent them. *Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... *Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. (The closer a phenomenon is to the prototype, the more likely it belong to the category). Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}} (see discussion here https://www.wikidata.org/wiki/Help:Basic_membership_properties). *Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}, {{P|1478}}, {{P|P9353}} (see discussions here : https://www.wikidata.org/wiki/Help:Modeling_causes/en). *Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can use it before really knowing what its properties will be. *Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Testing concept modelling on {{Wikidata entity link|Q14944319}} ==== To test concept modelling, we started by experimenting with the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic : *{{Wikidata entity link|Q137901202}} *{{Wikidata entity link|Q137901196}} *{{Wikidata entity link|Q137901182}} *{{Wikidata entity link|Q136447761}} *{{Wikidata entity link|Q129652515}} *{{Wikidata entity link|Q114306483}} We read each paper and used them as source in manually entered statements in the item {{Wikidata entity link|Q14944319}}. For example, Droubi et. Al stated "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}} and {{Wikidata entity link|Q3249551}}, using the paper as reference (see screenshot below). The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&oldid=2495982191. [[File:Wikidata statement- energy democracy is an instance of ideal.png|915x915px|border]] We listed the difficulties encountered as we worked and we also asked the Wikidata community to give us feedback on our modelling on the item discussion page (https://www.wikidata.org/wiki/Talk:Q14944319). These issues were related to contradictions, precision, concision and ontology. We discuss each issue and draft recommandations to refine our modelling process. ===== Contradictions ===== Wikidata contributor's feedback highlighted some apparent contradictions (The values in "does not have effect" seems contrary to what is listed in "has goal".) We would however argue this is not a problem because "statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata"<ref>{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}</ref>. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref>.Besides, in the case of goals versus effects statements, it is not contradictory because one can have a goal and fail to achieve it. In the case of energy democracy, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing<ref name=":20" />. Recommandations : Contradictions are allowed in Wikidata. ===== Precision ===== We noted that conceptual modelling requires an important degree of formalization and precision. This is a key advantage of Wikidata to be able to create links toward precise concepts which have their own identifiers. For example, we were able to create statements about specific laws and their unique identifiers in legal databases (e.g. {{Wikidata entity link|Q139764294}} and its identifier in the EUR-Lex database). However, the sources we are working with are not always precise enough and when concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, {{Wikidata entity link|Q12888920}}... We can see here that academic texts are using natural language and thus are using ambiguous terms. As a result, we received feedbacks regarding a lack of precision in our statements (too many and too vague statements). In addition to the ambiguity of sources, a reason why we ended up with very general statements is because we avoided the creation of new Wikidata items. While following this implicit rule allowed us to focus on the most notable concepts, creating new items could also help make the statements more precise. Recommandations : If a concept is ambiguous it should not be included in the modelled statements. Create more precise relevant concepts if they do not exist in Wikidata. [[File:Wikidata visualisation screenshot of subclasses relationships including the item political concept.png|thumb|298x298px|Subclass relationships between "concept" and "political concept".]] ===== Concision ===== Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were "in the tree of another value". The rule we take from this feeback is a need of logical simplification. Two examples illustrate possible logical simplification : *We stated that {{Wikidata entity link|Q14944319}} was an {{Wikidata entity link|P31}} {{Wikidata entity link|Q33104069}} and an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}. But in that case, it is not necessary to state that it is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}, because {{Wikidata entity link|Q33104069}} is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q131362181}}, which is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q151885}} (see diagram on the right). Here, we have to keep only the more precise item. This reasonning is based on the assumption that {{Wikidata entity link|P279}} is transitive. It seems this reasonning could be generalized (we opened a discussion about this here https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning/Use_cases#Parcimonious_statement_constraints_based_on_subclass_of_(P279)_and_part_of_(P361)_transitivity<nowiki/>) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using "instance of" as transversal property and checking "instance of or subclass of"). *We stated that {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P2670}} {{Wikidata entity link|Q15991216}} and {{Wikidata entity link|Q113514984}}. But if we consider that {{Wikidata entity link|Q15991216}} is a {{Wikidata entity link|P279}} of {{Wikidata entity link|Q113514984}}, then the inclusion of {{Wikidata entity link|Q15991216}} is implied. Here we could keep only the broader item {{Wikidata entity link|Q113514984}}. But unlike the reasonning above, we could not identify a way to generalize this reasonning. For example, {{Wikidata entity link|Q113514984}} can be a subclass of {{Wikidata entity link|Q43229}}, but using the former item is more informative than using the later. For the property {{Wikidata entity link|P2670}}, using a value that is too precise results in too many statements and using a value that is too general results in too trivial statements. Choosing the degree of precision is a problem similar to choosing whether or not to create a new item. Recommandations : When possible, use logical simplification to make statements more parcimonious. The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize redundancies. ===== Quantification ===== Modelling quantitative statements was challenging. We mostly skipped those but made an attempt for one case : the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources"<ref>{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research & Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}</ref>. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for "final energy consumption". There are such properties in Wikidata : for example, for renewable energy subsidies, there is {{Wikidata entity link|P6826}}. This type of property is notably useful to display information about an item (e.g. the population of a town) on Wikipedia through dynamic infoboxes that are updated with Wikidata information. But the current informal rule on Wikidata property creation is to prefer the use of general properties and avoid the creation of specific properties. Therefore creating properties for every quantifiable characteristic may not be possible. An alternative way to model quantity is to create a specific item to represent what is quantified and use other properties to represent quantities (see example below). [[File:Wikidata screenshot of percentage modelling.png|border|783x783px]] However, Wikidata is optimized for statements with a few discrete values and it is more convenient to store quantitative (tabular) data in elsewhere<ref>{{Cite journal|last=Larrañaga|first=Galder Gonzalez|last2=Pintscher|first2=Lydia|last3=Ainali|first3=Jan|last4=Fauconnier|first4=Sandra|date=July 2026|title=Tabular data vs structured data.pptx|url=https://docs.google.com/presentation/d/1k37UX-VPwMgJbXoB7ZxsztE5LdKR_isM|journal=Wikimania 2026}}</ref>. Instead, it might be more useful to link the item that is quantified to an external dataset about it. This can be done using the property {{Wikidata entity link|P1325}}. Recommandation : Use existing properties for the quantity if it exists. If it does not exist, create an item representing what is measured, link it toward an external datasets and add quantitative statements if relevant. ===== Ontology ===== Ultimately, several of our difficulties were linked to some ontology challenges reflecting the complexity of the concept we were working on. According to the litterature, {{Wikidata entity link|Q14944319}} refer to multiple types of entities. It represents a concept, an ideal, a process and an outcome. We reflected this with multiple {{Wikidata entity link|P31}} statements, but encountered struggles to model differences between ''process versus outcomes'', ''ideal versus reality'', ''phenomenon versus theory'' and ''discourses versus practices''. The wikidata community recommands having different items for ''process and outcome'' (criterion "{{Wikidata entity link|Q127270577}}"). For example, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork). However, this distinction is less straightforward for social processes which are are ongoing and evolving without a clear ending. Another problem was to separate ''ideal versus reality,'' we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes attempting to reach it. It was also difficult to separate ''phenomenon versus theory,'' for example, we did not manage to model the idea that the literature on energy democracy is fragmented. Finally, we also would have needed more distinctions between ''discourses and practices.'' Our sources suggested that energy democracy discourses and practices may have different causes, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movments promoting it, but this was not a very precise way to model this idea. Our first attempt to multiple "natures" of energy democracy into a single item is not satisfactory, and implementing the modelling recommandations we identified above may not be sufficient to build more parcimonious statements. Creating new items to reflect the different aspects of energy democracy may here be necessary. To do so we have to decide how to split the different natures {{Wikidata entity link|Q14944319}} and what will be the relationship between the resulting items. Here the definition of an {{Wikidata entity link|Q324254}} (formal representation) in Wikidata can quickly escalate into questionning {{Wikidata entity link|Q44325}} (metaphysical reflexion on the nature of things), and especially {{Wikidata entity link|Q1713511}} (the nature of the social world). Indeed, the relations between the different types of entities constituting the social worls are not simple nor consensual among philosophers of social science. For example, {{Wikidata entity link|Q15080858}} posits that different things have different ways of being (modes of reality). They propose to classify entities in four categories : material entities (that can exist independently of humans), conceptual entities (concepts, discourses, ideas, meaning…), artefactual entities (human-made and combining conceptual and material elements) and social entities (that depends on human activity to exist)<ref>Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.</ref>. A complex concept like {{Wikidata entity link|Q14944319}} is concerned with all these types of entities. The energy system include many material entities such as oil fields, the sun, seas, trees... and artefacts such as energy production unit, power lines, home appliances, trucks... (Including conceptual entities such as the name of these artefacts or the knowledge to make them function.). There are the social entities in which they are encompassed (the enregy sectors, energy businesses, energy policies...). There are conceptual entities like normative/political discourses discussing how these artefact and social system should work and there are conceptual entities in the academic sphere building theories about how all this works or should work. In theory, such ontology can be implemented in Wikidata as it "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8" /> Current Wikidata ontology is structured with dischotomies such as {{Wikidata entity link|Q7048977}}/{{Wikidata entity link|Q4406616}}, {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}, {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q23958946}}, {{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}, {{Wikidata entity link|Q30241068}}/{{Wikidata entity link|Q3799040}}... A quick search seem to show that entities related to the social reality can quickly fall into heterogeneous and sometime contradictory root classes (see table below). {| class="wikitable" |+ !Item !Existing root class entity |- |{{Wikidata entity link|Q190539}} |{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q67518978}} |- |{{Wikidata entity link|Q34770}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q9081}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q43229}} |{{Wikidata entity link|Q99527517}}{{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q103940464}} |- |{{Wikidata entity link|Q49773}} |{{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}} {{Wikidata entity link|Q99527517}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q12705}} |{{Wikidata entity link|Q4406616}}/{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}} {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q132907471}} |} Current {{Wikidata entity link|Q3882785}} in Wikidata do not seem very informative to classify the aspects of our concept in a top-down manner. We discussed with the Wikidata community<ref>{{Cite web|url=https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Ontology#Social_ontology|title=Wikidata talk:WikiProject Ontology - Wikidata|website=www.wikidata.org|language=en|access-date=2026-07-28}}</ref> and explored existing ontologies that could be relevant to our project. Some colleagues social science and humanities recommended CIDOC-CRM (https://cidoc-crm.org/crmsoc/fm_releases) because of the active community maintaining it. The conclusion of this exploration is that Wikidata, because of its collaborative nature, may not be the best plateform to implement a rigorous, interoperable, high-level ontology. We thus decided to focus on a lower ontological level. === Knowledge modelling in Wikidata : second round of analysis === Our first round of analysis resulted in : * Add links to exported files + insert graphs screenshots After our first round of analysis, we decided to focus on the strenghs of wikidata, which seem to be its ability to be a hub for concepts linked to external identifiers. We reframed our modelling goals to focus on identifying the main concepts in our corpus, their definition, their relationships, and their potential existing identifiers. Our next steps are * Read articles of the corpus again and model more precise statements * Include pages in references * Create new items to reflect the different facets of {{Wikidata entity link|Q14944319}} : the social movement advocating for it, the political concept theorizing it, the concrete initiatives implementing it, the litterature theorizing it, etc... based on a new reading of our sample of article and aligned with existing ontologies. == 3. Data visualisation == The goal of this step is to test '''Hypothesis 3''' (SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs). At this stage we started to identify visualization use cases we would like to test, they are listed below. === Filter statements === *Visualize only statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH *Visualize only items which are part of the present project (all the academic items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Properties visualisation === *Visualise the "tree" of a property used in an item : Wikidata graph builder seem to be the most user friendly, robust and versatile tool to visualise a graph of a single property (https://angryloki.github.io) === Concepts visualisation === *Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request "topic in context") *Map the statements that two items have in common to make comparisons. === Mapping sources consensus === *Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible. == 4. Writing == The goal of this step is to test '''Hypothesis 4''' (Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links). * To do idea : write a Wikiversity page for each central concept (e.g. energy democracy, energy community, energy citizenship...) === Advantages of Wikiversity === Writing on a Wikiversity page offers some advantages to implement the principles of open linked data in text format. We could cite academic items using their Wikidata QID to generate the citations below (but if we use an URL to cite a paper, it does not automatically check if there is an existing QID for it), and also link toward Wikidata entities using a template ([[Template:Wikidata entity link|Wikidata entity link]]). Although we did not translate the present page for now, Wikiversity offers the possibility to translate a page in multiple languages (an interesting possibility in the context of the The FAIRisation of scholarly communication<ref>{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}</ref>). The possibility to view the page history provide an exhaustive versionning of a paper, which constitute interesting data documenting the academic writing process (at first we did not comment the nature of our modifications but after thinking about this reuse possibility, we described it with more details). The contribution statistics based on the history also offers a new way to track author's contributions to a paper in a transparent way. Some researchers experimented relationships between the wikimedia community and researchers by making researchers review wikipedia articles<ref>{{Cite web|url=https://eprints.whiterose.ac.uk/id/eprint/226771/|title=CLOUDSENSE Collaborative Review of Wikipedia Cloud Feedback article|last=Finney|first=D.|last2=Huang|first2=X.|date=2025-05-16|website=eprints.whiterose.ac.uk|language=en|doi=10.48785/100/329|access-date=2026-07-28|last3=Hinnie|first3=L.|last4=Sheppard|first4=N.}}</ref>. === The issue of text interoperability === A key issue we are encountering is the question of the interoperability of texts. While the interoperability of data is starting to be well discussed in the open science community, the interoperability of texts do not seem to benefit from the same level of discussion. We encountered several interoperability issues regarding our writing. First, copying texts written on a word processor software (e.g. microsoft word) into a wiki page (or the other way around) is relatively seamless in terms of formatting, but the management of references is lacking. Reformatting references is very time consuming and a real barrier for text interoperability in academic context : it is difficult to copy text from an academic publication into a wiki text, and difficult to turn a wiki text into a publication (the only possibility is to export the page as a PDF and you may be reading this article in this format). There are also uncertaineties regarding how to reuse texts published under creative common licences. Academic texts published under CC-BY-SA licences can in theory be remixed and reused. But academia does not have established practices regarding how this can be done. If we want to reuse a whole page, should we put it in quotation marks and simply cite the paper ? Should the original authors be listed as co-authors ? To what extent academic publisher will accept to publish "remixed" texts while they usually require that publications contain mainly unpublished content ? The norms of what is appropriate remix and reuse practices in academia has yet to be decided... and this could be an interesting conversation to have in the open science community. == Discussion == Our goal was to assess the potential of Wikidata and Wikiversity to build a living literature review method and tackle issues of information overload, knowledge synthesis and knowledge dissemination, following open science principles. By conducting a meta-review on just sustainability transition (that is a review of existing literature reviews), and experimenting with existing technical solutions, we were able to identify the strenghs and limitations of the Wikidata ecosystem. First Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations (Hypothesis 1 is verified). However, the technical barriers to do so are still high and the Wikidata database is too general to allow a community to work on a specific curated corpus of scientific items. Secondly, Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (Hypothesis 2 is verified). However, this requires developping new methodological standards regarding what is a rigorous modelling in social science, as well as building specific {{Wikidata entity link|Q324254}}. An unexpected finding is that the formalization necessary for knowledge modelling in wikidata invites to formalize the reasonning behind modelling choices, a process that is rarely done in qualitative analysis even though such reasonning are likely present. The modelling process also responds to calls in management sciences to engage in more systematic categorizing to avoid semantic confusion<ref>{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}</ref>. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to "navigate" it (Hypothesis 3 is partially verified). SPARQL-based queries are powerful but they require technical knowledge, especially now that the split between academic and non-academic items in Wikidata requires to write federated queries. To be able to "navigate" scientific corpora and scientific knowledge graphs on Wikidata, more user-friendly tools would be needed. Finally, we did see advantaged in using Wikiversity pages to write collaboratively in text format augmented by hypertext links, but there are still important technical and instittional barriers (Hypothesis 3 is partially verified) : the interoperability of text is a key issue to reuse publications. The main issues we encountered were the management of references and uncertainty regarding the compatibilities between writing publicly on a wiki page and engaging in classical publication processes. A potential solution to the issues encountered could be to develop a specialised literature review software that would allow researchers to build living literature reviews including knowledge graphs and wiki pages. This kind of tool should be user-friendly and include the missing tools we identified. In addition, it could include more advanced analysis functions such as logical reasonning based on the knowledge graph<ref name=":9" />. === Future research === - Can we "reason" with Wikidata (See: https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning) : neurosymbolic AI ? - Compare grounded theory/CAQDAS with knowledge modelling methods, test annotation softwares : https://inception-project.github.io/ ; https://pacte.crim.ca/index_en.html == Funding == This project is funded by the [[m:Grants:Programs/Wikimedia_Research_&_Technology_Fund/Wikimedia_Research_Fund|Wikimedia Research Fund]], Grant ID: G-RS-2504-18935. The text of the initial research proposal is available here : https://doi.org/10.5281/zenodo.20760603. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 68n1i2j46hjtgjepj5bg52g9ihsvma3 2 326640 2819748 2818634 2026-07-29T11:29:25Z Atcovi 276019 /* June 2026 - August 2026 */ done 2819748 wikitext text/x-wiki == Application Goal == '''Fall 2027''' == Subpages == * [[User:Atcovi/Journey to Clinical PhD/Potential Clinical PhD programs]] * [[User:Atcovi/Journey to Clinical PhD/Research Interest?]] ** [[Suicide]] (WJ publication to improve [[w:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]]?) *[[User:Atcovi/Journey to Clinical PhD/Solid advice I've seen]] == Current tasks == === January 2026 - May 2026 === ==== Ongoing {{Ongoing}} ==== '''<u>High-priority</u>''' #Work on the CPARL Lab: [[User:Atcovi/Dr. Dzokoto/Emotion Lexicon Project]] {{Ongoing}} #Get ready for the call: [[User:Atcovi/Journey to Clinical PhD/feb19 zoom call|feb 19 zoom call info]] {{Done}} #Continue work on [[WikiJournal Preprints/Mental health in Sri Lanka]] [1] {{Done}} #start one on [[w:Overgeneral_autobiographical_memory|OGM]] [2]. {{Ongoing}} #*could this be qualitative? #**'''JULY 20, 2026:''' being worked off-wiki at the moment, hopefully will be published in september 2026. #Continue reaching out to PIs {{Ongoing}} '''<u>Less-priority</u>''' ---------------------------------------------------------------------------------------------------------------------------------------------- #[[User:Atcovi/Journey to Clinical PhD/Posters guidance|Try to go for posters/publications]] (writing committee [[User:Atcovi/Dr. Dzokoto|Dr. Dzokoto]]?). #* '''aim for 1-2 posters''' {{Ongoing}} #[[User:Atcovi/Journey to Clinical PhD/Researchers|Get a concrete list of researchers you'd like to work with and who are within your niche.]] {{Done}} #Attend MORE conferences (keep up with the American Association of Suicidology; virtual is valid as well). {{Ongoing}} #Secure a full-time, research-oriented job if you can. {{Ongoing}} #* [[User:Atcovi/Journey to Clinical PhD/tips for job search]] #Guest lecture? Adjunct professor? ==== Completed {{Completed}} ==== *Graduate college with a 4.0 GPA. {{Completed}} * Email [https://vipbg.vcu.edu/ Virginia Institute for Psychiatric and Behavioral Genetics] and try to get 1-2 professors to study under relating to suicidal ideation. {{Completed}} ** '''UPDATE:''' None available; probably due to emailing late, decent chance of getting a role in a year's time--important to grind in the meantime. *RE-SUBMIT [[Association Between Screen Time and Sleep: An Online Survey|undergrad thesis]] to PsyArXiv, then update WV page. {{Done}} ==== MAY 2026 PROGRESS REPORT: ==== 3 potential papers in the making, aim to get them published by fall 2026: # [[User:Atcovi/OGM & Suicide/The Paper]] # [[WikiJournal Preprints/Mental health in Sri Lanka]] {{Done}} # [[User:Atcovi/Dr. Dzokoto/Emotion Lexicon Project]] - Emotions paper 1 poster to present # [[User:Atcovi/APA2026 Abstract]] ← derived from paper #2. Continued internships? # Dr. Dzokoto (strengthen letter of recommendations) # '''EARLY AUGUST''' - reach out to professors at [https://vipbg.vcu.edu/ Virginia Institute for Psychiatric and Behavioral Genetics]. Continue applying everywhere. Job changes could change things dramatically. === June 2026 - August 2026 === * [[User:Atcovi/APA2026 Abstract]] - August 8th presentation * [[User:Atcovi/Journey to Clinical PhD/Dr. Nadorff Meeting]] - {{Done}} === September 2026 - December 2026 === ==== September 2026 PROGRESS REPORT: ==== [TBD] [[Category:Atcovi/Clinical Psychology]] m5ccvgb5lg7o65f44q6ac0gx5rx7vir 2 326765 2819692 2819310 2026-07-28T17:12:12Z Dc.samizdat 2856930 /* The 24-cell */ 2819692 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <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 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) 4-simplex, 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 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. Each isoclinic rotation takes two chiral forms. There is a right rotation and a left rotation for each row of the table. .... {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & 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}} d5y0fv7gxl15yw8mo0m9twydnpfaz8q 2819697 2819692 2026-07-28T18:16:09Z Dc.samizdat 2856930 /* The 24-cell */ 2819697 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <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 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) 4-simplex, 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 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, invariant as their vertices circle in 4-space. The isocline chords form the rotation's Clifford polygons, stationary in 4-space, over which the edge polygons' vertices circle. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. In the left rotation the roles of the edge chord polygon and the isocline chord polygon, as described above, are reversed: the edge chord polygon is the left rotation's stationary Clifford polygon over which the vertices circle, and the isocline chord polygon is the invariant configuration of the moving vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & 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}} 13paawufdbyalwjntswkuw22usl0ge1 2819699 2819697 2026-07-28T19:05:20Z Dc.samizdat 2856930 /* The 24-cell */ 2819699 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <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 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) 4-simplex, 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 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, over which vertices circle as the whole polygon moves orthogonally. The isocline chords form the rotation's Clifford polygons, which are stationary in 4-space as vertices circle over them. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. In the left rotation the roles of the edge chord polygons and the isocline chord polygons, as described above, are reversed: the edge chord polygon is stationary in 4-space as vertices circle over it, and the isocline chord polygon is an invariant configuration of moving vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & 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}} jvuca63tmfsv4ryr6nc55w0kj6aqvn3 2819700 2819699 2026-07-28T19:12:37Z Dc.samizdat 2856930 /* The 600-cell */ 2819700 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <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 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) 4-simplex, 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 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, over which vertices circle as the whole polygon moves orthogonally. The isocline chords form the rotation's Clifford polygons, which are stationary in 4-space as vertices circle over them. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. In the left rotation the roles of the edge chord polygon and the isocline chord polygon, as described above, are reversed: the edge chord polygon is stationary in 4-space as vertices circle over it, and the isocline chord polygon is an invariant configuration of moving vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & 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}} 1kgje2oz3y39oritruagarvu56z54dx 2819704 2819700 2026-07-28T19:31:50Z Dc.samizdat 2856930 /* The 600-cell */ 2819704 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <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 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) 4-simplex, 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 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 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). Consequently the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. Their procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-1</math> . Over the third <math>r_3</math> chord it also moves a distance of <math>-1</math>. Fontaine and Hurley also demonstrated the significance of <math>1/r_i</math> in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length <math>1/r_i</math> rather than <math>1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an <math>r_i</math> polygon. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. The angle between two <math>r_i</math> chords is <math>180^\circ - i \times 45^\circ</math>. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math> (it is <math>6\pi</math> in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes twelve 45° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the <math>r_3</math> {8/3} star polygon, which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a <math>\sqrt{3}</math> chord. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The <math>r_1</math> <small><math>\sqrt{1}</math></small> chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by <math>r_2</math> chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}. The <math>r_2</math> <small><math>\sqrt{1}</math></small> chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}. The <math>r_3</math> <small><math>\sqrt{2}</math></small> chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells. The <math>r_4</math> <small><math>\sqrt{3}</math></small> chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons. The <math>r_5</math> <small><math>\sqrt{3}</math></small> chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant <math>r_2</math> hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over <math>r_5</math> chords. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of <small><math>\sqrt{2}</math> </small>chords in the 24-cell]] We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° <small><math>\sqrt{2}</math></small> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <small><math>\sqrt{2}</math></small> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of <small><math>\sqrt{3}</math></small> chords. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of <small><math>\sqrt{3}</math></small> chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over <math>r_{5}</math> isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the <math>r_5</math> {12/5} star polygon which constructs <math>1/r_5</math>. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. An isoclinic rotation by 60° in an invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 invariant great hexagon planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 <math>r_5</math> <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each <math>r_5</math> 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |6 distinct 180° chord pairs make 6 distinct isoclinic rotations |- ! colspan="3" |Edge chords !Invariant planes ! colspan="3" |Isocline chords |- style="background: gainsboro;" | | rowspan="4" |<math>t_1</math> |60° | rowspan="4" |[[File:Regular_polygon_24.svg|100px]]<br>{24/1}={24} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-11.svg|100px]]<br>{24/11} |120° | rowspan="4" |<math>t_{11}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |165° |15° |- style="background: palegreen;" | | rowspan="4" |<math>t_2</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(12,1).svg|100px]]<br>{24/2}=2{12} | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="4" |<math>t_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |150° |30° |- style="background: seashell;" | | rowspan="4" |<math>t_3</math> |90° | rowspan="4" |[[File:Regular_star_figure_3(8,1).svg|100px]]<br>{24/3}=3{8} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_3(8,3).svg|100px]]<br>{24/9}=3{8/3} |90° | rowspan="4" |<math>t_{9}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |135° |45° |- style="background: palegreen;" | | rowspan="4" |<math>t_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_4(6,1).svg|100px]]<br>{24/4}=4{6} | rowspan="4" |[[File:Regular_star_figure_12(2,1).svg|100px]]<br>{24/12}=12{2} | rowspan="4" |[[File:Regular_star_figure_8(3,1).svg|100px]]<br>{24/8}=8{3} |120° | rowspan="4" |<math>t_{8}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro;" | | rowspan="4" |<math>t_5</math> |60° | rowspan="4" |[[File:Regular_star_polygon_24-5.svg|100px]]<br>{24/5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_24-7.svg|100px]]<br>{24/7} |120° | rowspan="4" |<math>t_{7}</math> |- style="background: gainsboro;" | |{{radic|1}} |{{radic|3}} |- style="background: gainsboro;" | |1 |1.732~ |- style="background: gainsboro;" | |105° |75° |- style="background: seashell;" | | rowspan="4" |<math>t_6</math> |90° | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} | rowspan="4" |[[File:Regular_star_figure_6(4,1).svg|100px]]<br>{24/6}=6{4} |90° | rowspan="4" |<math>t_{6}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} By examining the chords <math>r_i</math> of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords <math>t_i</math> of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations. Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. The short edge chord and long isocline chord each have their characteristic {24/n}-gon. The edge chords form the rotation's edge polygons, over which vertices circle as the whole polygon moves orthogonally. The isocline chords form the rotation's Clifford polygons, which are stationary in 4-space as vertices circle over them. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°. The invariant planes of the rotation intersect 0, 2, 4, or 6 vertices. Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. In the left rotation the roles of the edge chord polygon and the isocline chord polygon, as described above, are reversed: the edge chord polygon is stationary in 4-space as vertices circle over it, and the isocline chord polygon is an invariant configuration of moving vertices. {{Clear}} == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math> the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>r_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>r_1</math> |36° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="4" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>r_2</math> |36° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="4" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>r_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: palegreen;" | | rowspan="4" |<math>r_4</math> |60° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="4" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |132° |48° |- style="background: palegreen;" | | rowspan="4" |<math>r_5</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: yellow;" | | rowspan="4" |<math>r_{6}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: seashell;" | | rowspan="4" |<math>r_{7}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="4" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |96° |84° |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short edge chord and long isocline chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over <math>r_7</math> edge chords and <math>r_8</math> isocline chords. The {30/7} edge polygon is a skew helix of circumference <math>14\pi</math> with each <math>r_7</math> edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference <math>16\pi</math> with each <math>r_8</math> isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference <math>16\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>\sqrt{3}</math></small> ]] We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over <math>\sqrt{1}</math> edge chords and <math>\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over <math>r_{4}=\sqrt{1}</math> edge chords and <math>r_{11}=\sqrt{3}</math> isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each <math>r_{11}</math> isocline chord is the <math>\sqrt{3}</math> diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° <math>r_{11}</math> isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference <math>22\pi</math> over <math>r_{11}</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° <math>r_{3}</math> edges, over 144° <math>r_{12}</math> isocline chords. This ''great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each <math>r_{12}</math> chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions. The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° <math>r_{2}</math> edges, over 144° <math>r_{13}</math> isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° <math>r_{13}</math> isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference <math>26\pi</math> form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Edge chords ! Section ! colspan="3" |Isocline chords |- style="background: palegreen;" | | rowspan="4" |<math>c_0</math> |0° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="4" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | |180° |0° |- style="background: palegreen;" | | rowspan="4" |<math>c_1</math> |15.5~° | rowspan="4" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="4" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: palegreen;" | |168° |12° |- style="background: gainsboro;" | | rowspan="4" |<math>c_2</math> |25.2~° | rowspan="4" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="4" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: gainsboro;" | |156° |24° |- style="background: yellow;" | | rowspan="4" |<math>c_3</math> |36° | rowspan="4" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="4" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: yellow;" | |144° |36° |- style="background: gainsboro;" | | rowspan="4" |<math>c_4</math> |41.4~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |138.6~° | rowspan="4" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: gainsboro;" | |138° |42° |- style="background: palegreen;" | | rowspan="4" |<math>c_5</math> |44.5~° | rowspan="4" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="4" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: palegreen;" | |132° |48° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_6</math> |49.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |130.9~° | rowspan="4" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro;" | |128° |52° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_7</math> |56° | rowspan="4" | | rowspan="4" | | rowspan="4" | |124° | rowspan="4" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: gainsboro;" | |124° |56° |- style="background: palegreen;" | | rowspan="4" |<math>c_8</math> |60° | rowspan="4" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="4" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | |120° |60° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_9</math> |66.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |113.9~° | rowspan="4" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro;" | |116° |64° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{10}</math> |69.8~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |110.2~° | rowspan="4" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: gainsboro;" | |112° |68° |- style="background: yellow;" | | rowspan="4" |<math>c_{11}</math> |72° | rowspan="4" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="4" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: yellow;" | |108° |72° |- style="background: palegreen; height:50px" | | rowspan="4" |<math>c_{12}</math> |75.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="4" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: palegreen;" | |96° |84° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{13}</math> |81.1~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |98.9~° | rowspan="4" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro;" | |° |° |- style="background: gainsboro; height:50px" | | rowspan="4" |<math>c_{14}</math> |84.5~° | rowspan="4" | | rowspan="4" | | rowspan="4" | |95.5~° | rowspan="4" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: gainsboro;" | |° |° |- style="background: seashell;" | | rowspan="4" |<math>c_{15}</math> |90° | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="4" | | rowspan="4" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="4" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |- style="background: seashell;" | |90° |90° |} The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & 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}} kc32ewoxdtkediv4zd7oedplv2jryrw 4 327551 2819711 2816992 2026-07-28T20:16:35Z Codename Noreste 2969951 /* Classical guitar pedagogy */ archive from [[Wikiversity:Requests for Deletion]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2819711 wikitext text/x-wiki == [[Classical guitar pedagogy]] == {{archive top|'''Deleted''', per below [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:35, 16 July 2026 (UTC)}} According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC) :'''Weak delete''' This at least has <em>something</em> that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC) : '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:18, 23 May 2026 (UTC) : '''Delete''', the author is no longer here nor at Wikipedia, therefore it's unlikely to be developed for the foreseeable future. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:20, 10 July 2026 (UTC) :'''Delete''', no significant improvement in nearly two decades.{{diff|Classical guitar pedagogy|2810418|140826}}--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:38, 16 July 2026 (UTC) {{archive bottom}} ==[[Mippedia]] == {{archive top|Consensus to delete, and the author of the template did not respond to Jtneill's comment. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:37, 30 May 2026 (UTC)}} I propose the deletion of the page "[[Mippedia]]", due to the subject not being backed by reputable sources. Pages with the same subject has been deleted multiple times on the Indonesian Wikipedia. The original writer of the page did it solely to promote his wiki site. [[User:ANNAFscience|ANNAFscience]] ([[User talk:ANNAFscience|discuss]] • [[Special:Contributions/ANNAFscience|contribs]]) 10:39, 23 May 2026 (UTC) : {{ping|Sevent Me}} any comment? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:10, 23 May 2026 (UTC) :'''Delete''' I don't know what the point of this is. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:26, 23 May 2026 (UTC) : '''Delete'''. Advertising. Points to a non-English, copyright restricted website. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:58, 24 May 2026 (UTC) {{archive bottom}} ==[[United States UFO files]]== {{archive top|Deleted, but the author of the resource did not respond here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 31 May 2026 (UTC)}} Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:46, 21 May 2026 (UTC) :'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: "Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.")? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: "No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly."). :Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC) : Yes, could be speedy deleted. Otherwise, I don't know about the merits about leaving it around for 90 days, hence me bringing it to here. There is some comment in [[Wikiversity:Deletion policy]] about the specific deletion templates not being so important. More important I think is to flag for discussion. However, we could also improve the proposed policy to make the process clearer. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 21 May 2026 (UTC) : Ping {{u|User:Realcosmixyt}} for comment -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:54, 24 May 2026 (UTC) {{archive bottom}} ==Pages by Harold Foppele== {{archive top|All deleted, user page blanked [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:33, 6 June 2026 (UTC)}} [[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.: * [[Completing the square]] * [[Number of independent spatial modes in a spherical volume]] * [[Quantum]] ** [[Quantum/Andrew N. Jordan]] * [[Quantum A Matter Of Size]] * [[Quantum A Spooky Action at a Distance]] * [[Quantum: A Walk Through the Universe]] * [[Quantum Computing Algorithms in the NISQ Era]] * [[Quantum Formulas Collection]] * [[Quantum harmonic oscillator]] * [[Quantum Matter Elements and Particles]] * [[Quantum mechanics]] ** [[Quantum mechanics/Timeline]] * [[Quantum mechanics learning module]] * [[Quantum mechanics measurements]] * [[Quantum Noisy Qubits]] * [[Quantum optics beam splitter experiments]] * [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]] * [[Quantum Ultra fast lasers]] * [[Speed of sound experiments]] * [[User:Harold Foppele]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:12, 17 May 2026 (UTC) :'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:27, 17 May 2026 (UTC) {{Archive bottom}} == [[Film writing]] == {{archive top|Consensus to keep after vote changes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:20, 31 May 2026 (UTC)}} Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC) :<del>'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC)</del><ins>'''Keep''': It's now at least developed enough to be something. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:42, 31 May 2026 (UTC)</ins> : '''Keep''' as part of [[:Category:Film]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:57, 20 May 2026 (UTC) :@[[User:Atcovi|Atcovi]] @[[User:Koavf|Koavf]] The page seems to have been tidied up. Do you want to reevaluate your votes? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:40, 31 May 2026 (UTC) {{archive bottom}} == [[Emergency Operation Centre GIS]] == {{archive top|Consensus to delete. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:39, 30 May 2026 (UTC)}} Undeveloped for over a decade (only thing present is just an outline). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:44, 22 May 2026 (UTC) :*'''Delete''' :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:59, 22 May 2026 (UTC) :* '''Delete'''. Insufficiently developed. Was moved from [[b:Emergency Operation Centre GIS]]. : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:13, 23 May 2026 (UTC) {{archive bottom}} ==[[Wikiphilosophers]]== {{Archive top|Consensus is for '''deletion'''. Regardless, thank you for your efforts to improve Wikiversity @Perquirius. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:01, 27 June 2026 (UTC)}} Moving from {{tl|prod}} by {{at|Atcovi}}: "similar "philosophy"-related content has been removed in the past [issue of pseudoscience] + very little moderation (mirroring the issues of [[Wikidebates]]) + lacks educational value." The project has also been nominated for deletion on its talk page: [[Talk:Wikiphilosophers]]. There are many subpages: {{Special:PrefixIndex/Wikiphilosophers/}} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:45, 24 May 2026 (UTC) :'''Delete'''. Unfortunately, this project wasn't as successful as I had hoped. Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:29, 24 May 2026 (UTC) ::Don't forget to delete [[Template:Wikiphilosophers]], [[Template:Wikiphilosophers/doc]] and [[Template:Wikiphilosophers topics]] also. [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:30, 24 May 2026 (UTC) :'''Delete''' Wikiversity should not be used as a host for failed sister projects. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:06, 6 June 2026 (UTC) ::When will these pages be deleted? Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 12:13, 27 June 2026 (UTC) :::Don't forget to delete [[:Category:Wikiphilosophers]] and [[:Category:Wikiphilosophers templates]] also! Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 16:56, 27 June 2026 (UTC) {{archive bottom}} == [[Template:UserSkype]] == {{archive top|Deleted, template and category was removed from all pages. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:14, 14 June 2026 (UTC)}} Service was discontinued over a year ago. I suggest deleting the Userbox and [[:Category:Users familiar with Skype]], as it can only confuse or mislead. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:17, 30 May 2026 (UTC) :'''Delete''' per reasoning. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:40, 30 May 2026 (UTC) : '''Delete''' -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:48, 31 May 2026 (UTC) :Yup delete. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:59, 8 June 2026 (UTC) {{archive bottom}} q6cqquzse5joiy7xk7z50tpsvh4i3t4 2819713 2819711 2026-07-28T20:16:53Z Codename Noreste 2969951 /* Concomitant Strabismus */ archive from [[Wikiversity:Requests for Deletion]] ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2819713 wikitext text/x-wiki == [[Classical guitar pedagogy]] == {{archive top|'''Deleted''', per below [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:35, 16 July 2026 (UTC)}} According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC) :'''Weak delete''' This at least has <em>something</em> that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC) : '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:18, 23 May 2026 (UTC) : '''Delete''', the author is no longer here nor at Wikipedia, therefore it's unlikely to be developed for the foreseeable future. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:20, 10 July 2026 (UTC) :'''Delete''', no significant improvement in nearly two decades.{{diff|Classical guitar pedagogy|2810418|140826}}--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:38, 16 July 2026 (UTC) {{archive bottom}} == [[Concomitant Strabismus]] == {{archive top|Deleted per consensus. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:53, 16 July 2026 (UTC)}} Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC) :'''Delete''' per above. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:22, 12 July 2026 (UTC) {{archive bottom}} ==[[Mippedia]] == {{archive top|Consensus to delete, and the author of the template did not respond to Jtneill's comment. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:37, 30 May 2026 (UTC)}} I propose the deletion of the page "[[Mippedia]]", due to the subject not being backed by reputable sources. Pages with the same subject has been deleted multiple times on the Indonesian Wikipedia. The original writer of the page did it solely to promote his wiki site. [[User:ANNAFscience|ANNAFscience]] ([[User talk:ANNAFscience|discuss]] • [[Special:Contributions/ANNAFscience|contribs]]) 10:39, 23 May 2026 (UTC) : {{ping|Sevent Me}} any comment? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:10, 23 May 2026 (UTC) :'''Delete''' I don't know what the point of this is. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:26, 23 May 2026 (UTC) : '''Delete'''. Advertising. Points to a non-English, copyright restricted website. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:58, 24 May 2026 (UTC) {{archive bottom}} ==[[United States UFO files]]== {{archive top|Deleted, but the author of the resource did not respond here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:16, 31 May 2026 (UTC)}} Seems to be WP-like; material copied from [[w:United States UFO files]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 01:46, 21 May 2026 (UTC) :'''Delete''', but why would a PROD template not suffice? My logic was that it is a newly created page (made just today), and isn't a big project/difficult page to deal with. Do we not deal with newly created pages that appear to not satisfy Wikiversity's objectives/mission with a PROD template? Wouldn't we best reserve RFDs for long-standing pages (like the two pages above this section being listed for deletion) or ''after'' the PROD template isn't enough to determine the fate of such pages (per [[Wikiversity:Deletion policy#Proposed deletion (prod)|here]]: "Anyone still considering that the resource should be deleted [after the placement of the PROD template] may discuss deletion.")? A PROD template may also be useful in this case to alert the author that the page is not compatible with Wikiversity's learning objectives and communicates a concise opportunity to refine the page with the 90-day limit. Maybe even in this case, a speedy would've been enough (possibly fitting [[Wikiversity:Deletion policy#Criteria for speedy deletion|#12]]: "No research objectives or discussion in history. Welcome users and resources when likely to be expanded shortly."). :Interested to hear your thoughts as I want to make sure this is clear, as I've been cleaning up a lot of 'dead' pages around Wikiversity and find myself confused on whether to use PROD or RFD. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:08, 21 May 2026 (UTC) : Yes, could be speedy deleted. Otherwise, I don't know about the merits about leaving it around for 90 days, hence me bringing it to here. There is some comment in [[Wikiversity:Deletion policy]] about the specific deletion templates not being so important. More important I think is to flag for discussion. However, we could also improve the proposed policy to make the process clearer. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:20, 21 May 2026 (UTC) : Ping {{u|User:Realcosmixyt}} for comment -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:54, 24 May 2026 (UTC) {{archive bottom}} ==Pages by Harold Foppele== {{archive top|All deleted, user page blanked [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:33, 6 June 2026 (UTC)}} [[User:Harold Foppele]] is locally blocked indefinitely and globally banned for sockpuppetry. There were also WMF and local community concerns expressed about copyright violation and AI (over)use. As a result, I think the Wikiversity pages created by this account warrant review with regard what should be deleted, what should be retained etc.: * [[Completing the square]] * [[Number of independent spatial modes in a spherical volume]] * [[Quantum]] ** [[Quantum/Andrew N. Jordan]] * [[Quantum A Matter Of Size]] * [[Quantum A Spooky Action at a Distance]] * [[Quantum: A Walk Through the Universe]] * [[Quantum Computing Algorithms in the NISQ Era]] * [[Quantum Formulas Collection]] * [[Quantum harmonic oscillator]] * [[Quantum Matter Elements and Particles]] * [[Quantum mechanics]] ** [[Quantum mechanics/Timeline]] * [[Quantum mechanics learning module]] * [[Quantum mechanics measurements]] * [[Quantum Noisy Qubits]] * [[Quantum optics beam splitter experiments]] * [[Quantum: The Secret of Cohesion: How Waves Hold Matter Together]] * [[Quantum Ultra fast lasers]] * [[Speed of sound experiments]] * [[User:Harold Foppele]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:12, 17 May 2026 (UTC) :'''Delete all''' Not worth keeping. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:27, 17 May 2026 (UTC) {{Archive bottom}} == [[Film writing]] == {{archive top|Consensus to keep after vote changes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:20, 31 May 2026 (UTC)}} Undeveloped since 2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:05, 19 May 2026 (UTC) :<del>'''Delete''' Nothing here. Great idea in principle, tho. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:25, 20 May 2026 (UTC)</del><ins>'''Keep''': It's now at least developed enough to be something. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:42, 31 May 2026 (UTC)</ins> : '''Keep''' as part of [[:Category:Film]] resources. I've tidied the page, so it looks less abandoned. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:57, 20 May 2026 (UTC) :@[[User:Atcovi|Atcovi]] @[[User:Koavf|Koavf]] The page seems to have been tidied up. Do you want to reevaluate your votes? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:40, 31 May 2026 (UTC) {{archive bottom}} == [[Emergency Operation Centre GIS]] == {{archive top|Consensus to delete. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:39, 30 May 2026 (UTC)}} Undeveloped for over a decade (only thing present is just an outline). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:44, 22 May 2026 (UTC) :*'''Delete''' :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:59, 22 May 2026 (UTC) :* '''Delete'''. Insufficiently developed. Was moved from [[b:Emergency Operation Centre GIS]]. : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:13, 23 May 2026 (UTC) {{archive bottom}} ==[[Wikiphilosophers]]== {{Archive top|Consensus is for '''deletion'''. Regardless, thank you for your efforts to improve Wikiversity @Perquirius. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:01, 27 June 2026 (UTC)}} Moving from {{tl|prod}} by {{at|Atcovi}}: "similar "philosophy"-related content has been removed in the past [issue of pseudoscience] + very little moderation (mirroring the issues of [[Wikidebates]]) + lacks educational value." The project has also been nominated for deletion on its talk page: [[Talk:Wikiphilosophers]]. There are many subpages: {{Special:PrefixIndex/Wikiphilosophers/}} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 13:45, 24 May 2026 (UTC) :'''Delete'''. Unfortunately, this project wasn't as successful as I had hoped. Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:29, 24 May 2026 (UTC) ::Don't forget to delete [[Template:Wikiphilosophers]], [[Template:Wikiphilosophers/doc]] and [[Template:Wikiphilosophers topics]] also. [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 14:30, 24 May 2026 (UTC) :'''Delete''' Wikiversity should not be used as a host for failed sister projects. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 05:06, 6 June 2026 (UTC) ::When will these pages be deleted? Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 12:13, 27 June 2026 (UTC) :::Don't forget to delete [[:Category:Wikiphilosophers]] and [[:Category:Wikiphilosophers templates]] also! Kind regards, [[User:Perquirius|Perquirius]] ([[User talk:Perquirius|overleg]] • [[Special:Contributions/Perquirius|bijdragen]]) 16:56, 27 June 2026 (UTC) {{archive bottom}} == [[Template:UserSkype]] == {{archive top|Deleted, template and category was removed from all pages. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:14, 14 June 2026 (UTC)}} Service was discontinued over a year ago. I suggest deleting the Userbox and [[:Category:Users familiar with Skype]], as it can only confuse or mislead. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:17, 30 May 2026 (UTC) :'''Delete''' per reasoning. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:40, 30 May 2026 (UTC) : '''Delete''' -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:48, 31 May 2026 (UTC) :Yup delete. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:59, 8 June 2026 (UTC) {{archive bottom}} kpw3btmry65kum9tly8hkdxn2pwmzcc 2 328769 2819663 2809859 2026-07-28T15:06:04Z Atcovi 276019 Atcovi moved page [[Trirectangular Tetrahedrons]] to [[User:Tet-Math3/Trirectangular Tetrahedrons]] without leaving a redirect: low quality mathematics page/amateur mathematical crankery 2807435 wikitext text/x-wiki === <math>A\ Trirectangular\ Tetrahedron\ is\ a\ tetrahedron\ where\ the\ Origin\bigl(x,y,z\bigr)\ is\ composed\ of </math><math>3\ Right\ Angled\ Triangles.\ There\ are\ Five\ Hidden\ mathematical\ equalities\ in\ Trirectangular </math><math>Tetrahedrons.\ They\ exist\ only\ in\ these\ types\ of\ tetrahedrons. </math> === == . == === <big><math>Here\ are\ the\ Five\ Hidden\ Equalities\ and\ they\ are\ magical.</math></big> === === <big><math>K=\Biggl[\frac{\ a^2+b^2+c^2}{2}\Biggr]=\Bigl[x^2+y^2+z^2\Bigr]=\Bigl[a^2+z^2\Bigr]=\Bigl[b^2+y^2\Bigr]=\Bigl[c^2+x^2\Bigr]</math></big> === '''<big><math>If\ given\ \bigl(a,b,c\bigr)\ use\ the\ 1st\ term\ K=\frac{\ a^2+b^2+c^2}{2}\ \ This\ equation\ is\ a\ modification\ of </math></big>'''<big><math>Heron's\ Semiperimeter.\ Then\ use\ \bigl(a,b,c\bigr)\ to\ calculate\ \bigl(x,y,z\bigr)\ as\ shown\ beneath\ this\ diagram.</math></big>[[File:(ABC) = (XYZ).png|thumb|708x708px|CaliCat 1]] === <math>K\ equals\ all\ 5\ equations.</math> === === <big><math>K=\frac{\ a^2+b^2+c^2}{2}</math></big> === === <big><math>K=x^2+y^2+z^2</math></big> === === <big>''<math>K=a^2+z^2\qquad\ Blue</math>''</big> === === <big>''<math>K=b^2+y^2\qquad\ Black</math>''</big> === === <big>''<math>K=c^2+x^2\qquad\ Red</math>''</big> === ''.<big>And they interlock in such a way that they greatly simplify solving Trirectangular problems. Here the mating pairs are color coded for easy understanding.</big>'' <big>''The other way to remember which diagonal line'' (''a,b,c'') ''mates with it's rectangular counterpart'' (''x,y,z'') ''is to realize the mating pairs are the only lines that never touch each other'' !</big> <big><math>Thus\ A\ \ Mates\ only\ with\ Z\ \ Blue</math><math>\qquad\ \ B\ \ Mates\ only\ with\ Y\ \ Black</math><math>\qquad\ \ C\ \ Mates\ only\ with\ X\ \ Red</math></big> == .<math>IF\ \ given\ \ \bigl(a,b,c\bigr)\ \ then\ \ use\ \ K=\biggl[\frac{\ a^2+b^2+c^2}{2}\biggr]\ \ to\ calculate\ \ \bigl(x,y,z\bigr)</math><math>Then\ \ x=\sqrt{_{_.}K-c^2\ }\qquad\ y=\sqrt{_{_.}K-b^2\ }\qquad\ z=\sqrt{_{_.}K-a^2\ }</math> == === <big><math>IF\ \ Given\ \bigl(x,y,z\bigr)\ \ then\ \ use\ \ K=\Bigl[x^2+y^2+z^2\Bigr]\ \ to\ calculate\ \bigl(a,b,c\bigr)</math></big> === === <big><math>Then\quad a=\sqrt{_{_.}K-z^2\ }\qquad b=\sqrt{\ K-y^2\ }\qquad\ c=\sqrt{_{_.}K-x^2\ }</math></big> === === <big><math>Or:\quad\ a=\sqrt{_{_.}x^2+y^2\ }\qquad b=\sqrt{_{_.}x^2+z^2\ }\qquad c=\sqrt{_{_.}y^2+z^2\ }</math></big> === == . == === <big><math>Proof\ of\ the\ accuracy\ of\ the\ K\ formulas.\ Try\ these\ test\ numbers\ yourself.</math></big> === === <math>Given:\qquad a=14.4\qquad b=10\qquad c=12</math><big><math>Then\ use\ the\ Pythagorean\ Theorem\ to\ Calculate\ (x,y,z)</math></big> === === <math>K=\frac{\ a^2+b^2+c^2}{2}=225.68</math> === === <math>K=x^2+y^2+z^2\ \ =225.68</math><math>K=a^2+z^2\qquad\ \ \ =225.68\quad\bigl(a=14.4\quad z=4.280186911806539319650461\bigr)</math><math>K=b^2+y^2\qquad\ \ \ =225.68\quad\bigl(b=10\quad \ \ \ \ y=11.210709165793214885544255\bigr)</math><math>K=c^2+x^2\qquad\ \ \ =225.68\quad\bigl(c=12\quad \ \ \ \ x=9.037698822156002781875821\bigr)</math> === == . == === <math>The\ area\ of\ the\ Base\ Triangle\ derived\ by\ using\ \bigl(x,y,z\bigr)\ \&\ directly\ below\ that\ by\ using\ \bigl(a,b,c\bigr)</math><math>The\ \bigl(H\bigr)_{abc}\ designation\ is\ used\ to\ identify\ it\ as\ Heron's\ Triangle.\ \bigl(The\ BASE\ triangle\bigr)</math>. === === <math>Rectangular\ (x,y,z) \quad H_{abc}=\frac{\ \ \sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ }\ \ }{2_{.}}</math><math>Heron's\ Theorem\qquad H_{abc}=\sqrt{\ S(S-a)(S-b)(S-c)\ }\quad where\ \ S=\frac{\ a+b+c\ }{2}</math> === == . == === <math>Height\ of\ Tetrahedron\ from\ Base\ of\ Heron's\ Triangle\ H_{abc}\ to\ the\ origin\ \bigl(x,y,z\bigr)</math><math>This\ amazing\ equation\ was\ found\ on\ the\ internet.\ It\ is\ not\ my\ own\ work\ !</math><math>It\ is\ accurate\ and\ I\ Reworked\ it\ from\ its\ original\ form.\ \bigl(See\ Original\ Form\ at\ the\ bottom\ \bigr)</math>. === === ''<big><math>h=\frac{xyz}{\ \sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ \ }</math></big>'' <ref>{{Cite web|url=https://en.wikipedia.org/wiki/Trirectangular_tetrahedron|title=Trirectangular tetrahedron - Metric formulas}}</ref> <ref>{{Cite web|url=https://www.formuladen.com/en/|title=Formula Den}}</ref> === === <big><math>Areas:</math><math>A_{xy}=\frac{\ xy\ }{2}\qquad B_{xz}=\frac{\ xz\ }{2}\qquad C_{yz}=\frac{\ yz\ }{2}\qquad H_{abc}=\frac{\ \sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ }\ \ }{2}</math></big> === === <math>A_{total}=A_{xy}+B_{xz}+C_{yz}+H_{abc}</math> === === <big><math>A_{total}=\frac{\ xy+z(x+y)+\sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ }\ \ }{2}</math></big> === == . == === <big><math>Volume:</math></big> === === <math>V_{box}=xyz\qquad V_{tet}=\frac{\ V_{box}\ }{6}\qquad V_{tet}=\frac{\ xyz\ }{6}</math> === === . === <big><math>The\ similar\ appearence\ between\ Heron's\ semiperimeter\ \&\ its\ K\ substitute\ is\ obvious. </math><math>Heron's\ \ Theorem\qquad\ Trirectangular\ \ Theorem\ (?)</math></big> === <big><math>S=\frac{\ a+b+c\ }{2}\qquad K=\frac{\ a^2+b^2+c^2\ }{2}</math></big> === . == <math>Interesting\ Items</math> == === <big><math>K\ solutions\qquad\qquad\quad Transition\qquad\qquad\quad Perk\ solutions </math></big> === <big><math>x=\sqrt{_{_.}K-c^2\ }\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2+c^2\ }{2}-c^2\ }\qquad x=\sqrt{\ \frac{_{_.}a^2+b^2-c^2\ }{2}\ }</math></big> <big><math>y=\sqrt{_{_.}K-b^2\ }\qquad\ y=\sqrt{\ \frac{_{_.}a^2+b^2+c^2\ }{2}-b^2\ }\qquad y=\sqrt{\ \frac{_{_.}a^2+c^2-b^2\ }{2}\ }</math></big> <big><math>z=\sqrt{_{_.}K-a^2\ }\qquad z=\sqrt{\ \frac{_{_.}a^2+b^2+c^2\ }{2}-a^2\ }\qquad z=\sqrt{\ \frac{_{_.}b^2+c^2-a^2\ }{2}\ }</math></big> <big><math>It\ is\ interesting\ to\ note\ that\ in\ the\ perk\ column \ that\ all\ 3\ diagonals\ (a,b,c)\ are\ in\ the</math><math>numerators\ of\ each\ equation.\ And\ that\ any\ 2\ of\ the\ diagonals\ are\ added\ together\ while</math><math>the\ 3rd\ one\ is\ subtrated\ from\ that\ sum.\ And\ the\ solution\ is\ always\ the\ rectangular\ (x,y,z)</math><math>mate\ of\ the\ diagonal\ being\ subtracted.\ The\ mating\ pairs\ are\ the\ only\ lines\ that\ never\ touch</math><math>each\ other.</math></big> <big><math>1)\ \ Subtract\ (c)\ to\ get\ (x)\quad Because\ (c)\ never\ touches\ (x)</math><math>2)\ \ Subtract\ (b)\ to\ get\ (y)\quad Because\ (b)\ never\ touches\ (y)</math><math>3)\ \ Subtract\ (a)\ to\ get\ (z)\quad Because\ (a)\ never\ touches\ (z)</math></big> == <math>This\ amazing\ formula\ was\ found\ online\ BUT\ it\ is\ NOT\ my\ own \ work.</math> == <big><math>BUT\ no\ useable\ proof\ of\ its\ accuracy\ could\ be\ found\ online.</math></big> === <big><math>Original\ form\qquad \frac{1}{\ h^2}=\frac{1}{\ x^2}+\frac{1}{\ y^2}+\frac{1}{\ z^2}</math></big> === === <big><math>Transposed\ form\quad h=\frac{xyz}{\ \sqrt{\ x^2y^2+z^2\bigl(x^2+y^2\bigr)\ \ }\ \ }</math></big> === <big><math>The\ accuracy\ of\ \bigl(h\bigr)\ will\ be\ proven\ using\ Heron's\ Theorem.</math><math>The\ 1_{st}\ diagram\ CaliCat\ 1\ labels\ \bigl(x,y,z\bigr),\ \bigl (a,b,c\bigr)\ and\ \big(h\bigr)</math><math>The\ 2_{nd}\ diagram\ CaliCat\ 2\ labels\ \bigl(x,y,z\bigr),\bigl(x_{a},y_{a},z_{a}\bigr),\bigl(a,b,c\bigr)\ and\ \big(h\bigr)</math><math>The\ 3_{rd}\ diagram\ CaliCat\ 3\ labels\ \bigl(x,y,z\bigr),\bigl(x_{a},y_{a},z_{a}\bigr), \bigl(a,b,c\bigr)\ and\ \big(h\bigr)\ much\ more\ clearly.</math></big> <big><math>\bigl(h\bigr)\ is\ the\ green\ line\ between\ the\ origin\ \bigl(x,y,z\bigr)\ \&\ the\ point\ of\ intersection\ where\ \bigl(x_{a},y_{a},z_{a}\bigr)</math><math>meet\ on\ the\ \bigl(a,b,c\bigr)\ plain.\ This\ is\ the\ face\ of\ the\ base.</math></big> [[File:New Picture (8).png|thumb|490x490px|CaliCat 2]] . <big><math>x_{a}\ is\ the\ adjacent\ of\ \bigl(x,h\bigr)\quad x_{a}=\sqrt{_{.}x^2-h^2\ }</math></big> <big><math>y_{a}\ is\ the\ adjacent\ of\ \bigl(y,h\big)\ \quad y_{a}=\sqrt{_{.}y^2-h^2\ }</math></big> <big><math>z_{a}\ is\ the\ adjacent\ of\ \bigl(z,h\bigr)\quad\ z_{a}=\sqrt{_{.}z^2-h^2\ }</math></big> === <big><math>Heron's\ Theorem\ is\ applied\ 4\ times.</math></big> === <math>S=\frac{\ a+b+c\ }{2}\quad\quad\ H_{abc}=\sqrt{\ S \Bigl(S-a\Bigr)\Bigl(S-b\Bigr)\Bigl(S-c\Bigr)\ }</math> <math>S_{a}=\frac{\ a+x_{a}+y_{a}\ }{2}\quad A_{xy}=\sqrt{\ S_{a}\Bigl(S_{a}-a\Bigr)\Bigl(S_{a}-x_{a}\Bigr)\Bigl(S_{a}-y_{a}\Bigr)\ }</math> <math>S_{b}=\frac{\ b+x_{a}+z_{a}\ }{2}\ \quad B_{xz}=\sqrt{\ _{_.}S_{b}\Bigl(S_{b}-b\Bigr)\Bigl(S_{b}-x_{a}\Bigr)\Bigl(S_{b}-z_{a}\Bigr)\ }</math> <math>S_{c}=\frac{\ c+y_{a}+z_{a}\ }{2}\ \quad C_{yz}=\sqrt{\ _{_.}S_{c}\Bigl(S_{c}-c\Bigr)\Bigl(S_{c}-y_{a}\Bigr)\Bigl(S_{c}-z_{a}\Bigr)\ }</math> == . == <big><math>The\ purple\ triangle\ is\ the\ base\ facing\ down\ with\ the\ origin\ \bigl(x,y,z\bigr)\ pointing\ up.\ This\ is\ to</math><math>clearly\ show\ how\ the\ height\ \bigl(h\bigr)\ fits\ into\ the\ tetrahedron\ between\ the\ base\ \&\ the\ origin.</math></big> <big><math>Where:</math><math>X_{(Adjacent)}\ \ x_a=\sqrt{_{_.}x^2-h^2\ }\qquad Y_{(Adjacent)}\ \ y_a=\sqrt{_{_.}y^2-h^2\ }\qquad Z_{(Adjacent)}\ \ z_a =\sqrt{_{_.}z^2-h^2\ }</math></big> <big><math>This\ is\ an\ atempt\ to\ show\ why\ \bigl(h\bigr)\ must=\sqrt{\ x^2-{x_{a}}^2\ }=\sqrt{\ y^2-{y_{a}}^2\ }=\sqrt{\ z^2-{z_{a}}^2\ }</math></big> <big><math>Or\ the\ sum\ of\ the\ areas\ of\ triangles\ \bigl(a,x_{a},y_{a}\bigr)+\bigl(b,x_{a},z_{a}\bigr)+\bigl(c,y_{a},z_{a}\bigr)\ will\ not\ equal\ the </math><math>area\ of\ the\ base.\ Which\ is\ also\ Heron's\ triangle\ \bigl(a,b,c\bigr).\ \ Thus\ h=\frac{xyz}{\ 2A\ }\ where\ A=H_{abc}.</math></big> <big><math>Now\ it\ seems\ to\ be\ magical\ to\ know\ that\ the\ height\ \bigl(h\bigr)\ of\ a\ trirectangular\ tetrahedron</math><math>is\ simply\ the\ volume\ xyz\ divided\ by\ twice\ the\ base\ area.</math></big>[[File:New Picture (12).png|center|thumb|725x725px|CaliCat 3]] <big><math>Here\ the\ formulas\ are\ shown\ once\ again\ BUT\ directly\ under\ the\ clearer\ diagram\ \&\ with</math><math>the\ numbers\ that\ apply\ as\ proof\ of\ accuracy.</math></big> <big><math>S=\frac{\ a+b+c\ }{2}\quad\quad\ H_{abc}=\sqrt{\ S \Bigl(S-a\Bigr)\Bigl(S-b\Bigr)\Bigl(S-c\Bigr)\ }\qquad\qquad\ \ H_{abc}=59.296664324</math><math>S_{a}=\frac{\ a+x_{a}+y_{a}\ }{2}\quad A_{xy}=\sqrt{\ S_{a}\Bigl(S_{a}-a\Bigr)\Bigl(S_{a}-x_{a}\Bigr)\Bigl(S_{a}-y_{a}\Bigr)\ }\qquad A_{xy}=43.280437934</math></big><big><math>S_{b}=\frac{\ b+x_{a}+z_{a}\ }{2}\ \quad B_{xz}=\sqrt{\ _{_.}S_{b}\Bigl(S_{b}-b\Bigr)\Bigl(S_{b}-x_{a}\Bigr)\Bigl(S_{b}-z_{a}\Bigr)\ }\qquad\ B_{xz}=6.308860781</math><math>S_{c}=\frac{\ c+y_{a}+z_{a}\ }{2}\ \quad C_{yz}=\sqrt{\ _{_.}S_{c}\Bigl(S_{c}-c\Bigr)\Bigl(S_{c}-y_{a}\Bigr)\Bigl(S_{c}-z_{a}\Bigr)\ }\ \ \ \qquad C_{yz}=9.707365609</math></big> <big><math>If\ the\ value\ of\ \bigl(h\bigr)\ is\ correct\ then\ Heron's\ Area\ \bigl(A_{abc}\bigr)\ must\ equal\ the\ sum\ of\ the\ other\ 3. </math></big> <big><math>H_{abc}=59.296664324=A_{xy}+B_{xz}+C_{yz}=43.280437934\ +6.308860781\ +9.707365609</math></big> == . == <big><math>Original\ form\quad\frac{1}{\ h^2\ }=\frac{1}{\ x^2\ }+\frac{1}{\ y^2\ }+\frac{1}{\ z^2\ } </math></big> hg8vaudkzrmm6lbc5j3znblbn55ih94 3 328819 2819664 2802353 2026-07-28T15:06:04Z Atcovi 276019 Atcovi moved page [[Talk:Trirectangular Tetrahedrons]] to [[User talk:Tet-Math3/Trirectangular Tetrahedrons]] without leaving a redirect: low quality mathematics page/amateur mathematical crankery 2802353 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2 328820 2819747 2804390 2026-07-29T06:38:55Z Tet-Math3 3062596 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletion policy|deleted]]. 2819747 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 0 329548 2819701 2819635 2026-07-28T19:17:18Z Athene241 3100061 /* Solve the problem */ 2819701 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484; or for prime bases ''b'': https://oeis.org/A122396), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored). The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]]. To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''^''n''+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1 we only used it to sieve the sequence ''a''×''b''^''n''+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''^''n''+''c'', but 2 may not divide (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the "easy" bases (bases ''b'' with ≤ 150 primes > 10²⁹⁹ (base ''b'' = 26 has 83 known primes > 10²⁹⁹ and 3 unsolved families, base ''b'' = 36 has 75 known primes > 10²⁹⁹ and 4 unsolved families, base ''b'' = 17 has 99 known primes > 10²⁹⁹ and 18 unsolved families, base ''b'' = 21 has 80 known primes > 10²⁹⁹ and 12 unsolved families, base ''b'' = 19 has 201 known primes > 10²⁹⁹ and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation. == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == Condensed table for bases 2 ≤ ''b'' ≤ 36 == {|class="wikitable" ||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write "''d''_''n''" if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)|| |- ||2||1||11||2||1||3||http://factordb.com/index.php?id=3&open=ecm||http://factordb.com/index.php?showid=3&base=2||0||–|| |- ||3||3||111<br>21<br>12||3<br>2<br>2||2<br>1<br>1||13<br>7<br>5||http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=13&base=3<br>http://factordb.com/index.php?showid=7&base=3<br>http://factordb.com/index.php?showid=5&base=3<nowiki/>||0||–|| |- ||4||5||221<br>31<br>23<br>13<br>11||3<br>2<br>2<br>2<br>2||2<br>2<br>2<br>1<br>1||41<br>13<br>11<br>7<br>5||http://factordb.com/index.php?id=41&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=41&base=4<br>http://factordb.com/index.php?showid=13&base=4<br>http://factordb.com/index.php?showid=11&base=4<br>http://factordb.com/index.php?showid=7&base=4<br>http://factordb.com/index.php?showid=5&base=4<nowiki/>||0||–|| |- ||5||22||10₉₃13<br>300031<br>44441<br>33331<br>33001<br>30301<br>14444<br>10103<br>3101<br>414||96<br>6<br>5<br>5<br>5<br>5<br>5<br>5<br>4<br>3||67<br>4<br>4<br>4<br>4<br>4<br>4<br>3<br>3<br>3||5⁹⁵+8<br>9391<br>3121<br>2341<br>2251<br>1951<br>1249<br>653<br>401<br>109||http://factordb.com/index.php?id=1100000000034686071&open=ecm<br>http://factordb.com/index.php?id=9391&open=ecm<br>http://factordb.com/index.php?id=3121&open=ecm<br>http://factordb.com/index.php?id=2341&open=ecm<br>http://factordb.com/index.php?id=2251&open=ecm<br>http://factordb.com/index.php?id=1951&open=ecm<br>http://factordb.com/index.php?id=1249&open=ecm<br>http://factordb.com/index.php?id=653&open=ecm<br>http://factordb.com/index.php?id=401&open=ecm<br>http://factordb.com/index.php?id=109&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000034686071&base=5<br>http://factordb.com/index.php?showid=9391&base=5<br>http://factordb.com/index.php?showid=3121&base=5<br>http://factordb.com/index.php?showid=2341&base=5<br>http://factordb.com/index.php?showid=2251&base=5<br>http://factordb.com/index.php?showid=1951&base=5<br>http://factordb.com/index.php?showid=1249&base=5<br>http://factordb.com/index.php?showid=653&base=5<br>http://factordb.com/index.php?showid=401&base=5<br>http://factordb.com/index.php?showid=109&base=5<nowiki/>||0||–|| |- ||6||11||40041<br>4441<br>4401<br>51<br>45<br>35<br>31<br>25<br>21<br>15||5<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||4<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||5209<br>1033<br>1009<br>31<br>29<br>23<br>19<br>17<br>13<br>11||http://factordb.com/index.php?id=5209&open=ecm<br>http://factordb.com/index.php?id=1033&open=ecm<br>http://factordb.com/index.php?id=1009&open=ecm<br>http://factordb.com/index.php?id=31&open=ecm<br>http://factordb.com/index.php?id=29&open=ecm<br>http://factordb.com/index.php?id=23&open=ecm<br>http://factordb.com/index.php?id=19&open=ecm<br>http://factordb.com/index.php?id=17&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<nowiki/>||http://factordb.com/index.php?showid=5209&base=6<br>http://factordb.com/index.php?showid=1033&base=6<br>http://factordb.com/index.php?showid=1009&base=6<br>http://factordb.com/index.php?showid=31&base=6<br>http://factordb.com/index.php?showid=29&base=6<br>http://factordb.com/index.php?showid=23&base=6<br>http://factordb.com/index.php?showid=19&base=6<br>http://factordb.com/index.php?showid=17&base=6<br>http://factordb.com/index.php?showid=13&base=6<br>http://factordb.com/index.php?showid=11&base=6<nowiki/>||0||–|| |- ||7||71||3₁₆1<br>510₇1<br>3₆01<br>1100021<br>531101<br>351101<br>300053<br>150001<br>100121<br>40054||17<br>10<br>8<br>7<br>6<br>6<br>6<br>6<br>6<br>5||15<br>9<br>7<br>6<br>5<br>5<br>5<br>5<br>5<br>4||(7¹⁷−5)/2<br>36×7⁸+1<br>(7⁸−47)/2<br>134471<br>91631<br>62819<br>50459<br>28813<br>16871<br>9643||http://factordb.com/index.php?id=116315256993601&open=ecm<br>http://factordb.com/index.php?id=207532837&open=ecm<br>http://factordb.com/index.php?id=2882377&open=ecm<br>http://factordb.com/index.php?id=134471&open=ecm<br>http://factordb.com/index.php?id=91631&open=ecm<br>http://factordb.com/index.php?id=62819&open=ecm<br>http://factordb.com/index.php?id=50459&open=ecm<br>http://factordb.com/index.php?id=28813&open=ecm<br>http://factordb.com/index.php?id=16871&open=ecm<br>http://factordb.com/index.php?id=9643&open=ecm<nowiki/>||http://factordb.com/index.php?showid=116315256993601&base=7<br>http://factordb.com/index.php?showid=207532837&base=7<br>http://factordb.com/index.php?showid=2882377&base=7<br>http://factordb.com/index.php?showid=134471&base=7<br>http://factordb.com/index.php?showid=91631&base=7<br>http://factordb.com/index.php?showid=62819&base=7<br>http://factordb.com/index.php?showid=50459&base=7<br>http://factordb.com/index.php?showid=28813&base=7<br>http://factordb.com/index.php?showid=16871&base=7<br>http://factordb.com/index.php?showid=9643&base=7<nowiki/>||0||–|| |- ||8||75||4₂₂₀7<br>5₁₃25<br>7₁₂1<br>77774₆1<br>74₇1<br>4₈1<br>5₅025<br>5550525<br>5500525<br>4₅77||221<br>15<br>13<br>11<br>9<br>9<br>8<br>7<br>7<br>7||200<br>14<br>12<br>10<br>9<br>8<br>8<br>7<br>7<br>7||(4×8²²¹+17)/7<br>(5×8¹⁵−173)/7<br>8¹³−7<br>(28669×8⁷−25)/7<br>(53×8⁸−25)/7<br>(4×8⁹−25)/7<br>(5×8⁸−2413)/7<br>1495381<br>1474901<br>(4×8⁷+185)/7||http://factordb.com/index.php?id=1100000000416605822&open=ecm<br>http://factordb.com/index.php?id=25131694349141&open=ecm<br>http://factordb.com/index.php?id=549755813881&open=ecm<br>http://factordb.com/index.php?id=8589035809&open=ecm<br>http://factordb.com/index.php?id=127027489&open=ecm<br>http://factordb.com/index.php?id=76695841&open=ecm<br>http://factordb.com/index.php?id=11983381&open=ecm<br>http://factordb.com/index.php?id=1495381&open=ecm<br>http://factordb.com/index.php?id=1474901&open=ecm<br>http://factordb.com/index.php?id=1198399&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000416605822&base=8<br>http://factordb.com/index.php?showid=25131694349141&base=8<br>http://factordb.com/index.php?showid=549755813881&base=8<br>http://factordb.com/index.php?showid=8589035809&base=8<br>http://factordb.com/index.php?showid=127027489&base=8<br>http://factordb.com/index.php?showid=76695841&base=8<br>http://factordb.com/index.php?showid=11983381&base=8<br>http://factordb.com/index.php?showid=1495381&base=8<br>http://factordb.com/index.php?showid=1474901&base=8<br>http://factordb.com/index.php?showid=1198399&base=8<nowiki/>||0||–|| |- ||9||151||30₁₁₅₈11<br>27₆₈₆07<br>76₃₂₉2<br>561₃₆<br>10₂₅57<br>30₂₀51<br>8₁₉335<br>727₁₅07<br>51₁₃61<br>10₁₁507||1161<br>689<br>331<br>38<br>28<br>23<br>22<br>19<br>16<br>15||1108<br>657<br>316<br>37<br>26<br>22<br>21<br>19<br>16<br>14||3×9¹¹⁶⁰+10<br>(23×9⁶⁸⁸−511)/8<br>(31×9³³⁰−19)/4<br>(409×9³⁶−1)/8<br>9²⁷+52<br>3×9²²+46<br>9²²−454<br>(527×9¹⁷−511)/8<br>(41×9¹⁵+359)/8<br>9¹⁴+412||http://factordb.com/index.php?id=1100000002376318423&open=prime<br>http://factordb.com/index.php?id=1100000002495467486&open=prime<br>http://factordb.com/index.php?id=1100000002359003642&open=prime<br>http://factordb.com/index.php?id=1100000001554010824&open=ecm<br>http://factordb.com/index.php?id=1100000002512830927&open=ecm<br>http://factordb.com/index.php?id=1100000000032261811&open=ecm<br>http://factordb.com/index.php?id=1100000002495736583&open=ecm<br>http://factordb.com/index.php?id=1100000003446800389&open=ecm<br>http://factordb.com/index.php?id=1055192051985121&open=ecm<br>http://factordb.com/index.php?id=22876792455373&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002376318423&base=9<br>http://factordb.com/index.php?showid=1100000002495467486&base=9<br>http://factordb.com/index.php?showid=1100000002359003642&base=9<br>http://factordb.com/index.php?showid=1100000001554010824&base=9<br>http://factordb.com/index.php?showid=1100000002512830927&base=9<br>http://factordb.com/index.php?showid=1100000000032261811&base=9<br>http://factordb.com/index.php?showid=1100000002495736583&base=9<br>http://factordb.com/index.php?showid=1100000003446800389&base=9<br>http://factordb.com/index.php?showid=1055192051985121&base=9<br>http://factordb.com/index.php?showid=22876792455373&base=9<nowiki/>||0||–|| |- ||10||77||50₂₈27<br>5₁₁1<br>805₅1<br>66600049<br>66000049<br>60₅49<br>220₅1<br>5200007<br>946669<br>666649||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||5×10³⁰+27<br>(5×10¹²−41)/9<br>(725×10⁶−41)/9<br>66600049<br>66000049<br>6×10⁷+49<br>22×10⁶+1<br>5200007<br>946669<br>666649||http://factordb.com/index.php?id=1100000000204142046&open=ecm<br>http://factordb.com/index.php?id=555555555551&open=ecm<br>http://factordb.com/index.php?id=80555551&open=ecm<br>http://factordb.com/index.php?id=66600049&open=ecm<br>http://factordb.com/index.php?id=66000049&open=ecm<br>http://factordb.com/index.php?id=60000049&open=ecm<br>http://factordb.com/index.php?id=22000001&open=ecm<br>http://factordb.com/index.php?id=5200007&open=ecm<br>http://factordb.com/index.php?id=946669&open=ecm<br>http://factordb.com/index.php?id=666649&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000204142046&base=10<br>http://factordb.com/index.php?showid=555555555551&base=10<br>http://factordb.com/index.php?showid=80555551&base=10<br>http://factordb.com/index.php?showid=66600049&base=10<br>http://factordb.com/index.php?showid=66000049&base=10<br>http://factordb.com/index.php?showid=60000049&base=10<br>http://factordb.com/index.php?showid=22000001&base=10<br>http://factordb.com/index.php?showid=5200007&base=10<br>http://factordb.com/index.php?showid=946669&base=10<br>http://factordb.com/index.php?showid=666649&base=10<nowiki/>||0||–|| |- ||11||1068||57₆₂₆₆₈<br>557₁₀₁₁<br>7₇₅₉44<br>A₇₁₃58<br>85₂₂₀05<br>507₂₀₆<br>5₁₆₁2A<br>50₁₂₆57<br>10₁₂₅51<br>326₁₂₂||62669<br>1013<br>761<br>715<br>223<br>208<br>163<br>129<br>128<br>124||65263<br>1055<br>793<br>745<br>233<br>217<br>170<br>134<br>133<br>129||(57×11⁶²⁶⁶⁸−7)/10<br>(607×11¹⁰¹¹−7)/10<br>(7×11⁷⁶¹−367)/10<br>11⁷¹⁵−58<br>(17×11²²²−111)/2<br>(557×11²⁰⁶−7)/10<br>(11¹⁶³−57)/2<br>5×11¹²⁸+62<br>11¹²⁷+56<br>(178×11¹²²−3)/5||http://factordb.com/index.php?id=1100000003573679860&open=prime<br>http://factordb.com/index.php?id=1100000002361376522&open=prime<br>http://factordb.com/index.php?id=1100000002505568840&open=prime<br>http://factordb.com/index.php?id=1100000003576826487&open=prime<br>http://factordb.com/index.php?id=1100000003576826769&open=ecm<br>http://factordb.com/index.php?id=1100000002518512744&open=ecm<br>http://factordb.com/index.php?id=1100000002391585327&open=ecm<br>http://factordb.com/index.php?id=1100000002632393378&open=ecm<br>http://factordb.com/index.php?id=1100000002391531300&open=ecm<br>http://factordb.com/index.php?id=1100000003576826781&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003573679860&base=11<br>http://factordb.com/index.php?showid=1100000002361376522&base=11<br>http://factordb.com/index.php?showid=1100000002505568840&base=11<br>http://factordb.com/index.php?showid=1100000003576826487&base=11<br>http://factordb.com/index.php?showid=1100000003576826769&base=11<br>http://factordb.com/index.php?showid=1100000002518512744&base=11<br>http://factordb.com/index.php?showid=1100000002391585327&base=11<br>http://factordb.com/index.php?showid=1100000002632393378&base=11<br>http://factordb.com/index.php?showid=1100000002391531300&base=11<br>http://factordb.com/index.php?showid=1100000003576826781&base=11<nowiki/>||0||–|| |- ||12||106||40₃₉77<br>B0₂₇9B<br>B₆99B<br>AA0₅1<br>B00099B<br>AAA0001<br>BBBAA1<br>A00065<br>44AAA1<br>BBBB1||42<br>30<br>9<br>8<br>7<br>7<br>6<br>6<br>6<br>5||45<br>33<br>10<br>9<br>8<br>8<br>7<br>7<br>7<br>6||4×12⁴¹+91<br>11×12²⁹+119<br>12⁹−313<br>130×12⁶+1<br>32847239<br>32555521<br>2985817<br>2488397<br>1097113<br>248821||http://factordb.com/index.php?id=1100000002375054575&open=ecm<br>http://factordb.com/index.php?id=1100000002354113100&open=ecm<br>http://factordb.com/index.php?id=5159780039&open=ecm<br>http://factordb.com/index.php?id=388177921&open=ecm<br>http://factordb.com/index.php?id=32847239&open=ecm<br>http://factordb.com/index.php?id=32555521&open=ecm<br>http://factordb.com/index.php?id=2985817&open=ecm<br>http://factordb.com/index.php?id=2488397&open=ecm<br>http://factordb.com/index.php?id=1097113&open=ecm<br>http://factordb.com/index.php?id=248821&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002375054575&base=12<br>http://factordb.com/index.php?showid=1100000002354113100&base=12<br>http://factordb.com/index.php?showid=5159780039&base=12<br>http://factordb.com/index.php?showid=388177921&base=12<br>http://factordb.com/index.php?showid=32847239&base=12<br>http://factordb.com/index.php?showid=32555521&base=12<br>http://factordb.com/index.php?showid=2985817&base=12<br>http://factordb.com/index.php?showid=2488397&base=12<br>http://factordb.com/index.php?showid=1097113&base=12<br>http://factordb.com/index.php?showid=248821&base=12<nowiki/>||0||–|| |- ||13||3197||A3₅₉₂₁₉₇A<br>95₁₉₇₄₂₀<br>80₃₂₀₁₇111<br>C5₂₃₇₅₅C<br>C₁₀₆₃₁92<br>B0₆₅₄₀BBA<br>390₆₂₆₆1<br>1770₂₇₀₃17<br>720₂₂₉₇2<br>930₁₅₅₁1||592199<br>197421<br>32021<br>23757<br>10633<br>6544<br>6269<br>2708<br>2300<br>1554||659677<br>219916<br>35670<br>26464<br>11845<br>7290<br>6983<br>3016<br>2562<br>1731||(41×13⁵⁹²¹⁹⁸+27)/4<br>(113×13¹⁹⁷⁴²⁰−5)/12<br>8×13³²⁰²⁰+183<br>(149×13²³⁷⁵⁶+79)/12<br>13¹⁰⁶³³−50<br>11×13⁶⁵⁴³+2012<br>48×13⁶²⁶⁷+1<br>267×13²⁷⁰⁵+20<br>93×13²²⁹⁸+2<br>120×13¹⁵⁵²+1||http://factordb.com/index.php?id=1100000005489162806&open=prime<br>http://factordb.com/index.php?id=1100000003943359311&open=prime<br>http://factordb.com/index.php?id=1100000000490878060&open=prime<br>http://factordb.com/index.php?id=1100000003590647776&open=prime<br>http://factordb.com/index.php?id=1100000003590493750&open=prime<br>http://factordb.com/index.php?id=1100000002616382906&open=prime<br>http://factordb.com/index.php?id=1100000000765961441&open=prime<br>http://factordb.com/index.php?id=1100000003590430825&open=prime<br>http://factordb.com/index.php?id=1100000002632396910&open=prime<br>http://factordb.com/index.php?id=1100000000765961452&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005489162806&base=13<br>http://factordb.com/index.php?showid=1100000003943359311&base=13<br>http://factordb.com/index.php?showid=1100000000490878060&base=13<br>http://factordb.com/index.php?showid=1100000003590647776&base=13<br>http://factordb.com/index.php?showid=1100000003590493750&base=13<br>http://factordb.com/index.php?showid=1100000002616382906&base=13<br>http://factordb.com/index.php?showid=1100000000765961441&base=13<br>http://factordb.com/index.php?showid=1100000003590430825&base=13<br>http://factordb.com/index.php?showid=1100000002632396910&base=13<br>http://factordb.com/index.php?showid=1100000000765961452&base=13<nowiki/>||0||–|| |- ||14||650||4D₁₉₆₉₈<br>34D₇₀₈<br>8D₁₄₁85<br>8₈₆B<br>40₈₃49<br>8C₇₉3<br>18₇₉B<br>6B₇₇2B<br>4₆₃09<br>A₅₉3||19699<br>710<br>144<br>87<br>86<br>81<br>81<br>80<br>65<br>60||22578<br>814<br>165<br>100<br>99<br>93<br>92<br>92<br>74<br>69||5×14¹⁹⁶⁹⁸−1<br>47×14⁷⁰⁸−1<br>9×14¹⁴³−79<br>(8×14⁸⁷+31)/13<br>4×14⁸⁵+65<br>(116×14⁸⁰−129)/13<br>(21×14⁸⁰+31)/13<br>(89×14⁷⁹−1649)/13<br>(4×14⁶⁵−667)/13<br>(10×14⁶⁰−101)/13||http://factordb.com/index.php?id=1100000000884560233&open=prime<br>http://factordb.com/index.php?id=1100000001540144903&open=prime<br>http://factordb.com/index.php?id=1100000003575856650&open=ecm<br>http://factordb.com/index.php?id=1100000002321014379&open=ecm<br>http://factordb.com/index.php?id=1100000000823937973&open=ecm<br>http://factordb.com/index.php?id=1100000002631073246&open=ecm<br>http://factordb.com/index.php?id=1100000002384401372&open=ecm<br>http://factordb.com/index.php?id=1100000002631077787&open=ecm<br>http://factordb.com/index.php?id=1100000000840126683&open=ecm<br>http://factordb.com/index.php?id=1100000002321038522&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000884560233&base=14<br>http://factordb.com/index.php?showid=1100000001540144903&base=14<br>http://factordb.com/index.php?showid=1100000003575856650&base=14<br>http://factordb.com/index.php?showid=1100000002321014379&base=14<br>http://factordb.com/index.php?showid=1100000000823937973&base=14<br>http://factordb.com/index.php?showid=1100000002631073246&base=14<br>http://factordb.com/index.php?showid=1100000002384401372&base=14<br>http://factordb.com/index.php?showid=1100000002631077787&base=14<br>http://factordb.com/index.php?showid=1100000000840126683&base=14<br>http://factordb.com/index.php?showid=1100000002321038522&base=14<nowiki/>||0||–|| |- ||15||1284||7₁₅₅97<br>E₁₄₅397<br>96₁₀₄08<br>7₇₃CE<br>7₅₉CCE<br>50₃₃17<br>EB₃₁<br>6330₂₆1<br>7050₂₄B<br>B70₂₄1||157<br>148<br>107<br>75<br>62<br>36<br>32<br>30<br>28<br>27||185<br>175<br>126<br>88<br>73<br>42<br>38<br>35<br>33<br>32||(15¹⁵⁷+59)/2<br>15¹⁴⁸−2558<br>(66×15¹⁰⁶−619)/7<br>(15⁷⁵+163)/2<br>(15⁶²+2413)/2<br>5×15³⁵+22<br>(207×15³¹−11)/14<br>1398×15²⁷+1<br>1580×15²⁵+11<br>172×15²⁵+1||http://factordb.com/index.php?id=1100000002454891840&open=ecm<br>http://factordb.com/index.php?id=1100000002454900849&open=ecm<br>http://factordb.com/index.php?id=1100000000823937997&open=ecm<br>http://factordb.com/index.php?id=1100000003588407143&open=ecm<br>http://factordb.com/index.php?id=1100000003588407386&open=ecm<br>http://factordb.com/index.php?id=1100000002632398579&open=ecm<br>http://factordb.com/index.php?id=1100000002321033312&open=ecm<br>http://factordb.com/index.php?id=1100000002391199877&open=ecm<br>http://factordb.com/index.php?id=1100000003588407806&open=ecm<br>http://factordb.com/index.php?id=1100000000851967288&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002454891840&base=15<br>http://factordb.com/index.php?showid=1100000002454900849&base=15<br>http://factordb.com/index.php?showid=1100000000823937997&base=15<br>http://factordb.com/index.php?showid=1100000003588407143&base=15<br>http://factordb.com/index.php?showid=1100000003588407386&base=15<br>http://factordb.com/index.php?showid=1100000002632398579&base=15<br>http://factordb.com/index.php?showid=1100000002321033312&base=15<br>http://factordb.com/index.php?showid=1100000002391199877&base=15<br>http://factordb.com/index.php?showid=1100000003588407806&base=15<br>http://factordb.com/index.php?showid=1100000000851967288&base=15<nowiki/>||0||–|| |- ||16||2347||3₁₁₆₁₃₇AF<br>4₇₂₇₈₅DD<br>DB₃₂₂₃₄<br>D0B₁₇₈₀₄<br>5BC₃₇₀₀D<br>90₃₅₄₂91<br>300F₁₉₆₀AF<br>20₁₇₁₃321<br>F8₁₅₁₇F<br>FAF₁₀₆₂45||116139<br>72787<br>32235<br>17806<br>3703<br>3545<br>1965<br>1717<br>1519<br>1066||139845<br>87644<br>38815<br>21441<br>4459<br>4269<br>2366<br>2067<br>1830<br>1284||(16¹¹⁶¹³⁹+619)/5<br>(4×16⁷²⁷⁸⁷+2291)/15<br>(206×16³²²³⁴−11)/15<br>(3131×16¹⁷⁸⁰⁴−11)/15<br>(459×16³⁷⁰¹+1)/5<br>9×16³⁵⁴⁴+145<br>769×16¹⁹⁶²−81<br>2×16¹⁷¹⁶+801<br>(233×16¹⁵¹⁸+97)/15<br>251×16¹⁰⁶⁴−187||http://factordb.com/index.php?id=1100000003851731988&open=prime<br>http://factordb.com/index.php?id=1100000003615909841&open=prime<br>http://factordb.com/index.php?id=1100000002383583629&open=prime<br>http://factordb.com/index.php?id=1100000003589278511&open=prime<br>http://factordb.com/index.php?id=1100000000993764322&open=prime<br>http://factordb.com/index.php?id=1100000000633424191&open=prime<br>http://factordb.com/index.php?id=1100000003588368750&open=prime<br>http://factordb.com/index.php?id=1100000003588386735&open=prime<br>http://factordb.com/index.php?id=1100000000633744824&open=prime<br>http://factordb.com/index.php?id=1100000003588387610&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003851731988&base=16<br>http://factordb.com/index.php?showid=1100000003615909841&base=16<br>http://factordb.com/index.php?showid=1100000002383583629&base=16<br>http://factordb.com/index.php?showid=1100000003589278511&base=16<br>http://factordb.com/index.php?showid=1100000000993764322&base=16<br>http://factordb.com/index.php?showid=1100000000633424191&base=16<br>http://factordb.com/index.php?showid=1100000003588368750&base=16<br>http://factordb.com/index.php?showid=1100000003588386735&base=16<br>http://factordb.com/index.php?showid=1100000000633744824&base=16<br>http://factordb.com/index.php?showid=1100000003588387610&base=16<nowiki/>||0||–|| |- ||17||10415~10427||95F₁₉₈₈₅₅<br>B0₁₈₉₀₈₃DB<br>F70₁₈₆₇₆₇1<br>970₁₆₆₀₄₇1<br>510₁₂₄₀₇₄D<br>49₁₁₁₃₃₃<br>B₆₇₁₀₃2E<br>570₅₁₃₁₀1<br>E9B₄₄₇₃₂<br>D0GD₃₇₀₉₆||198857<br>189086<br>186770<br>166050<br>124077<br>111334<br>67105<br>51313<br>44734<br>37099||244684<br>232661<br>229811<br>204316<br>152670<br>136991<br>82570<br>63138<br>55043<br>45649||(2543×17¹⁹⁸⁸⁵⁵−15)/16<br>11×17¹⁸⁹⁰⁸⁵+232<br>262×17¹⁸⁶⁷⁶⁸+1<br>160×17¹⁶⁶⁰⁴⁸+1<br>86×17¹²⁴⁰⁷⁵+13<br>(73×17¹¹¹³³³−9)/16<br>(11×17⁶⁷¹⁰⁵−2411)/16<br>92×17⁵¹³¹¹+1<br>(3963×17⁴⁴⁷³²−11)/16<br>(60381×17³⁷⁰⁹⁶−13)/16||http://factordb.com/index.php?id=1100000008610514108&open=prime<br>http://factordb.com/index.php?id=1100000008610515753&open=prime<br>http://factordb.com/index.php?id=1100000000765961429&open=prime<br>http://factordb.com/index.php?id=1100000000765961411&open=prime<br>http://factordb.com/index.php?id=1100000008610516879&open=prime<br>http://factordb.com/index.php?id=1100000000808118219&open=prime<br>http://factordb.com/index.php?id=1100000003993647842&open=prime<br>http://factordb.com/index.php?id=1100000000765961389&open=prime<br>http://factordb.com/index.php?id=1100000003883765450&open=prime<br>http://factordb.com/index.php?id=1100000003848346668&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008610514108&base=17<br>http://factordb.com/index.php?showid=1100000008610515753&base=17<br>http://factordb.com/index.php?showid=1100000000765961429&base=17<br>http://factordb.com/index.php?showid=1100000000765961411&base=17<br>http://factordb.com/index.php?showid=1100000008610516879&base=17<br>http://factordb.com/index.php?showid=1100000000808118219&base=17<br>http://factordb.com/index.php?showid=1100000003993647842&base=17<br>http://factordb.com/index.php?showid=1100000000765961389&base=17<br>http://factordb.com/index.php?showid=1100000003883765450&base=17<br>http://factordb.com/index.php?showid=1100000003848346668&base=17<nowiki/>||12||200000|| |- ||18||549||C0₆₂₆₈C5<br>H₇₆₆FH<br>80₂₉₈B<br>C0₁₁₆F5<br>HD₉₃<br>GG0₃₀1<br>CF₃₀5<br>B₁₉6B<br>CCF₁₄5<br>7₁₄G7||6271<br>768<br>300<br>119<br>94<br>33<br>32<br>21<br>17<br>16||7872<br>965<br>377<br>150<br>118<br>42<br>41<br>27<br>22<br>20||12×18⁶²⁷⁰+221<br>18⁷⁶⁸−37<br>8×18²⁹⁹+11<br>12×18¹¹⁸+275<br>(302×18⁹³−13)/17<br>304×18³¹+1<br>(219×18³¹−185)/17<br>(11×18²¹−1541)/17<br>(3891×18¹⁵−185)/17<br>(7×18¹⁶+2747)/17||http://factordb.com/index.php?id=1100000003590442437&open=prime<br>http://factordb.com/index.php?id=1100000003590430490&open=prime<br>http://factordb.com/index.php?id=1100000002355574745&open=prime<br>http://factordb.com/index.php?id=1100000002632837015&open=ecm<br>http://factordb.com/index.php?id=1100000002321052894&open=ecm<br>http://factordb.com/index.php?id=1100000000819230161&open=ecm<br>http://factordb.com/index.php?id=1100000002631240657&open=ecm<br>http://factordb.com/index.php?id=1100000003590430474&open=ecm<br>http://factordb.com/index.php?id=1100000003590430470&open=ecm<br>http://factordb.com/index.php?id=1100000003590430465&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003590442437&base=18<br>http://factordb.com/index.php?showid=1100000003590430490&base=18<br>http://factordb.com/index.php?showid=1100000002355574745&base=18<br>http://factordb.com/index.php?showid=1100000002632837015&base=18<br>http://factordb.com/index.php?showid=1100000002321052894&base=18<br>http://factordb.com/index.php?showid=1100000000819230161&base=18<br>http://factordb.com/index.php?showid=1100000002631240657&base=18<br>http://factordb.com/index.php?showid=1100000003590430474&base=18<br>http://factordb.com/index.php?showid=1100000003590430470&base=18<br>http://factordb.com/index.php?showid=1100000003590430465&base=18<nowiki/>||0||–|| |- ||19||31417~31434||1E70₁₂₂₈₉₆1<br>40₁₂₁₈₄₆HB5<br>35₁₂₀₅₆₂<br>FH0H₁₁₂₆₅₉<br>FG6₁₁₀₉₈₄<br>H₈₆₂₉₁6<br>D90₇₃₀₄₆9<br>4F0₄₉₈₄₇6<br>2₄₈₂₂₄7<br>2₄₅₈₈₆7A||122900<br>121850<br>120563<br>112662<br>110986<br>86292<br>73049<br>49850<br>48225<br>45888||157158<br>155816<br>154170<br>144067<br>110347<br>141924<br>93412<br>63746<br>61667<br>58679||634×19¹²²⁸⁹⁷+1<br>4×19¹²¹⁸⁴⁹+6351<br>(59×19¹²⁰⁵⁶²−5)/18<br>(103301×19¹¹²⁶⁵⁹−17)/18<br>(904×19¹¹⁰⁹⁸⁴−1)/3<br>(17×19⁸⁶²⁹²−215)/18<br>256×19⁷³⁰⁴⁷+9<br>91×19⁴⁹⁸⁴⁸+6<br>(19⁴⁸²²⁵+44)/9<br>(19⁴⁵⁸⁸⁸+926)/9||http://factordb.com/index.php?id=1100000001582289581&open=prime<br>http://factordb.com/index.php?id=1100000008755307222&open=prime<br>http://factordb.com/index.php?id=1100000005513825027&open=prime<br>http://factordb.com/index.php?id=1100000008755311453&open=prime<br>http://factordb.com/index.php?id=1100000000808118212&open=prime<br>http://factordb.com/index.php?id=1100000004163040839&open=prime<br>http://factordb.com/index.php?id=1100000003998413751&open=prime<br>http://factordb.com/index.php?id=1100000000808118332&open=prime<br>http://factordb.com/index.php?id=1100000003949188041&open=prime<br>http://factordb.com/index.php?id=1100000003949189035&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000001582289581&base=19<br>http://factordb.com/index.php?showid=1100000008755307222&base=19<br>http://factordb.com/index.php?showid=1100000005513825027&base=19<br>http://factordb.com/index.php?showid=1100000008755311453&base=19<br>http://factordb.com/index.php?showid=1100000000808118212&base=19<br>http://factordb.com/index.php?showid=1100000004163040839&base=19<br>http://factordb.com/index.php?showid=1100000003998413751&base=19<br>http://factordb.com/index.php?showid=1100000000808118332&base=19<br>http://factordb.com/index.php?showid=1100000003949188041&base=19<br>http://factordb.com/index.php?showid=1100000003949189035&base=19<nowiki/>||17||200000|| |- ||20||3314||G0₆₂₆₉D<br>CD₂₄₄₉<br>50₁₁₆₃AJ<br>J₆₅₅05J<br>JCJ₆₂₉<br>E₅₆₆C7<br>3A₅₂₇3<br>G₄₄₇99<br>EC0₄₂₉7<br>40₃₈₇404B||6271<br>2450<br>1166<br>658<br>631<br>568<br>529<br>449<br>432<br>392||8159<br>3188<br>1517<br>857<br>821<br>739<br>688<br>585<br>562<br>510||16×20⁶²⁷⁰+13<br>(241×20²⁴⁴⁹−13)/19<br>5×20¹¹⁶⁵+219<br>20⁶⁵⁸−7881<br>393×20⁶²⁹−1<br>(14×20⁵⁶⁸−907)/19<br>(67×20⁵²⁸−143)/19<br>(16×20⁴⁴⁹−2809)/19<br>292×20⁴³⁰+7<br>4×20³⁹¹+32091||http://factordb.com/index.php?id=1100000003590539457&open=prime<br>http://factordb.com/index.php?id=1100000002325393915&open=prime<br>http://factordb.com/index.php?id=1100000003590502412&open=prime<br>http://factordb.com/index.php?id=1100000003590502490&open=prime<br>http://factordb.com/index.php?id=1100000001559454258&open=prime<br>http://factordb.com/index.php?id=1100000003590502516&open=prime<br>http://factordb.com/index.php?id=1100000003590502531&open=prime<br>http://factordb.com/index.php?id=1100000000840126753&open=prime<br>http://factordb.com/index.php?id=1100000002633348702&open=prime<br>http://factordb.com/index.php?id=1100000003590502563&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003590539457&base=20<br>http://factordb.com/index.php?showid=1100000002325393915&base=20<br>http://factordb.com/index.php?showid=1100000003590502412&base=20<br>http://factordb.com/index.php?showid=1100000003590502490&base=20<br>http://factordb.com/index.php?showid=1100000001559454258&base=20<br>http://factordb.com/index.php?showid=1100000003590502516&base=20<br>http://factordb.com/index.php?showid=1100000003590502531&base=20<br>http://factordb.com/index.php?showid=1100000000840126753&base=20<br>http://factordb.com/index.php?showid=1100000002633348702&base=20<br>http://factordb.com/index.php?showid=1100000003590502563&base=20<nowiki/>||0||–|| |- ||21||13386~13394||27₁₈₄₄₉₉9D<br>F9₁₇₈₇₇₁D<br>2FC₁₁₂₀₂₂A<br>7₁₀₈₄₅₀ID<br>40₄₇₃₃₃9G<br>B90₄₅₀₁₉E5<br>HD₃₇₄₁₄<br>BD₃₅₀₂₇B<br>990₃₃₂₃₉99H<br>5₃₀₆₀₆FEK||184502<br>178773<br>112025<br>108452<br>47336<br>45023<br>37415<br>35029<br>33244<br>30609||243952<br>236377<br>148121<br>143397<br>62588<br>59531<br>49471<br>46316<br>43956<br>40472||(47×21¹⁸⁴⁵⁰¹+953)/20<br>(309×21¹⁷⁸⁷⁷²+71)/20<br>(288×21¹¹²⁰²³−13)/5<br>(7×21¹⁰⁸⁴⁵²+4733)/20<br>4×21⁴⁷³³⁵+205<br>240×21⁴⁵⁰²¹+299<br>(353×21³⁷⁴¹⁴−13)/20<br>(233×21³⁵⁰²⁸−53)/20<br>198×21³³²⁴²+4175<br>(21³⁰⁶⁰⁹+18455)/4||http://factordb.com/index.php?id=1100000008700600990&open=prime<br>http://factordb.com/index.php?id=1100000008700596669&open=prime<br>http://factordb.com/index.php?id=1100000008700593358&open=prime<br>http://factordb.com/index.php?id=1100000008700586183&open=prime<br>http://factordb.com/index.php?id=1100000000808118331&open=prime<br>http://factordb.com/index.php?id=1100000003996110311&open=prime<br>http://factordb.com/index.php?id=1100000003996110479&open=prime<br>http://factordb.com/index.php?id=1100000003996110718&open=prime<br>http://factordb.com/index.php?id=1100000003996110944&open=prime<br>http://factordb.com/index.php?id=1100000003996111130&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008700600990&base=21<br>http://factordb.com/index.php?showid=1100000008700596669&base=21<br>http://factordb.com/index.php?showid=1100000008700593358&base=21<br>http://factordb.com/index.php?showid=1100000008700586183&base=21<br>http://factordb.com/index.php?showid=1100000000808118331&base=21<br>http://factordb.com/index.php?showid=1100000003996110311&base=21<br>http://factordb.com/index.php?showid=1100000003996110479&base=21<br>http://factordb.com/index.php?showid=1100000003996110718&base=21<br>http://factordb.com/index.php?showid=1100000003996110944&base=21<br>http://factordb.com/index.php?showid=1100000003996111130&base=21<nowiki/>||8||200000|| |- ||22||8003||BK₂₂₀₀₁5<br>7₃₈₁₅2L<br>L₂₃₈₅KE7<br>7₉₅₉K7<br>J0₇₆₇IGGJ<br>K0₇₆₀EC1<br>I₆₂₆AF<br>E60₄₉₆L<br>L₄₈₃G3<br>L0₄₅₄B63||22003<br>3817<br>2388<br>961<br>772<br>764<br>628<br>499<br>485<br>458||29538<br>5124<br>3206<br>1290<br>1037<br>1026<br>843<br>670<br>652<br>615||(251×22²²⁰⁰²−335)/21<br>(22³⁸¹⁷−289)/3<br>22²³⁸⁸−653<br>(22⁹⁶¹+857)/3<br>19×22⁷⁷¹+199779<br>20×22⁷⁶³+7041<br>(6×22⁶²⁸−1259)/7<br>314×22⁴⁹⁷+21<br>22⁴⁸⁵−129<br>21×22⁴⁵⁷+5459||http://factordb.com/index.php?id=1100000003594696838&open=prime<br>http://factordb.com/index.php?id=1100000003591359839&open=prime<br>http://factordb.com/index.php?id=1100000003591360774&open=prime<br>http://factordb.com/index.php?id=1100000003591361817&open=prime<br>http://factordb.com/index.php?id=1100000003591362567&open=prime<br>http://factordb.com/index.php?id=1100000000632724415&open=prime<br>http://factordb.com/index.php?id=1100000000632724334&open=prime<br>http://factordb.com/index.php?id=1100000000632703239&open=prime<br>http://factordb.com/index.php?id=1100000003591364730&open=prime<br>http://factordb.com/index.php?id=1100000003591365331&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003594696838&base=22<br>http://factordb.com/index.php?showid=1100000003591359839&base=22<br>http://factordb.com/index.php?showid=1100000003591360774&base=22<br>http://factordb.com/index.php?showid=1100000003591361817&base=22<br>http://factordb.com/index.php?showid=1100000003591362567&base=22<br>http://factordb.com/index.php?showid=1100000000632724415&base=22<br>http://factordb.com/index.php?showid=1100000000632724334&base=22<br>http://factordb.com/index.php?showid=1100000000632703239&base=22<br>http://factordb.com/index.php?showid=1100000003591364730&base=22<br>http://factordb.com/index.php?showid=1100000003591365331&base=22<nowiki/>||0||–|| |- ||23||65178~65265||B0₉₃₀₄₆FB<br>L₈₆₄₄₄D<br>AJ₈₁₀₆₅4<br>20₇₃₅₆₀98<br>J₆₈₂₁₇G4<br>D70₆₆₇₇₀B<br>5F₆₂₃₄₀6<br>A7M7₆₁₅₃₂<br>B30₆₁₁₃₆5<br>EJ₅₂₁₆₉||93049<br>86445<br>81067<br>73563<br>68219<br>66773<br>62342<br>61535<br>61139<br>52170||126708<br>117715<br>110391<br>100172<br>92896<br>90927<br>84893<br>83794<br>83255<br>71042||11×23⁹³⁰⁴⁸+356<br>(21×23⁸⁶⁴⁴⁵−197)/22<br>(239×23⁸¹⁰⁶⁶−349)/22<br>2×23⁷³⁵⁶²+215<br>(19×23⁶⁸²¹⁹−1867)/22<br>306×23⁶⁶⁷⁷¹+11<br>(125×23⁶²³⁴¹−213)/22<br>(120413×23⁶¹⁵³²−7)/22<br>256×23⁶¹¹³⁷+5<br>(327×23⁵²¹⁶⁹−19)/22||http://factordb.com/index.php?id=1100000004691540361&open=prime<br>http://factordb.com/index.php?id=1100000004691546739&open=prime<br>http://factordb.com/index.php?id=1100000004691548070&open=prime<br>http://factordb.com/index.php?id=1100000004691548569&open=prime<br>http://factordb.com/index.php?id=1100000004691549462&open=prime<br>http://factordb.com/index.php?id=1100000004691549803&open=prime<br>http://factordb.com/index.php?id=1100000004691551005&open=prime<br>http://factordb.com/index.php?id=1100000004691556967&open=prime<br>http://factordb.com/index.php?id=1100000004691557254&open=prime<br>http://factordb.com/index.php?id=1100000004691557548&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004691540361&base=23<br>http://factordb.com/index.php?showid=1100000004691546739&base=23<br>http://factordb.com/index.php?showid=1100000004691548070&base=23<br>http://factordb.com/index.php?showid=1100000004691548569&base=23<br>http://factordb.com/index.php?showid=1100000004691549462&base=23<br>http://factordb.com/index.php?showid=1100000004691549803&base=23<br>http://factordb.com/index.php?showid=1100000004691551005&base=23<br>http://factordb.com/index.php?showid=1100000004691556967&base=23<br>http://factordb.com/index.php?showid=1100000004691557254&base=23<br>http://factordb.com/index.php?showid=1100000004691557548&base=23<nowiki/>||87||100000|| |- ||24||3409||N00N₈₁₂₉LN<br>88N₅₉₅₁<br>A0₂₉₅₁8ID<br>D₂₆₉₈LD<br>N₂₆₄₄LLN<br>BC0₃₃₁B<br>20₃₁₃7<br>C7₂₉₈<br>D0₂₅₉KKD<br>I0₂₄₁I5||8134<br>5953<br>2955<br>2700<br>2647<br>334<br>315<br>299<br>263<br>244||11227<br>8216<br>4079<br>3727<br>3654<br>461<br>434<br>413<br>363<br>337||13249×24⁸¹³¹−49<br>201×24⁵⁹⁵¹−1<br>10×24²⁹⁵⁴+5053<br>(13×24²⁷⁰⁰+4403)/23<br>24²⁶⁴⁷−1201<br>276×24³³²+11<br>2×24³¹⁴+7<br>(283×24²⁹⁸−7)/23<br>13×24²⁶²+12013<br>18×24²⁴³+437||http://factordb.com/index.php?id=1100000003593391606&open=prime<br>http://factordb.com/index.php?id=1100000003593275880&open=prime<br>http://factordb.com/index.php?id=1100000003593269654&open=prime<br>http://factordb.com/index.php?id=1100000003593269876&open=prime<br>http://factordb.com/index.php?id=1100000003593270089&open=prime<br>http://factordb.com/index.php?id=1100000002633359842&open=prime<br>http://factordb.com/index.php?id=1100000002355610241&open=prime<br>http://factordb.com/index.php?id=1100000002326181235&open=prime<br>http://factordb.com/index.php?id=1100000003593270725&open=prime<br>http://factordb.com/index.php?id=1100000002633360037&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003593391606&base=24<br>http://factordb.com/index.php?showid=1100000003593275880&base=24<br>http://factordb.com/index.php?showid=1100000003593269654&base=24<br>http://factordb.com/index.php?showid=1100000003593269876&base=24<br>http://factordb.com/index.php?showid=1100000003593270089&base=24<br>http://factordb.com/index.php?showid=1100000002633359842&base=24<br>http://factordb.com/index.php?showid=1100000002355610241&base=24<br>http://factordb.com/index.php?showid=1100000002326181235&base=24<br>http://factordb.com/index.php?showid=1100000003593270725&base=24<br>http://factordb.com/index.php?showid=1100000002633360037&base=24<nowiki/>||0||–|| |- ||25||133639~133724||E₉₈₃₉₆FOO<br>1J710₉₆₂₇₂1<br>NB0₈₅₅₉₈5NH<br>D70₈₁₅₈₁JJ7<br>F0₈₀₀₅₄HL<br>J010₇₅₉₄₃E7<br>K₆₇₇₇₁5I<br>LO₆₆₃₇₇KC<br>KJD0₆₃₃₉₉1<br>70₆₀₈₉₂D711||98399<br>96277<br>85603<br>81586<br>80057<br>75948<br>67773<br>66380<br>63403<br>60897||137556<br>134589<br>119668<br>114053<br>111915<br>106171<br>94743<br>92796<br>88634<br>85130||(7×25⁹⁸³⁹⁹+10613)/12<br>27676×25⁹⁶²⁷³+1<br>586×25⁸⁵⁶⁰¹+3717<br>332×25⁸¹⁵⁸⁴+12357<br>15×25⁸⁰⁰⁵⁶+446<br>11876×25⁷⁵⁹⁴⁵+357<br>(5×25⁶⁷⁷⁷³−2267)/6<br>22×25⁶⁶³⁷⁹−113<br>12988×25⁶³⁴⁰⁰+1<br>7×25⁶⁰⁸⁹⁶+207526||http://factordb.com/index.php?id=1100000000808118215&open=prime<br>http://factordb.com/index.php?id=1100000003983674902&open=prime<br>http://factordb.com/index.php?id=1100000004909706420&open=prime<br>http://factordb.com/index.php?id=1100000004909733266&open=prime<br>http://factordb.com/index.php?id=1100000004909750102&open=prime<br>http://factordb.com/index.php?id=1100000004909770736&open=prime<br>http://factordb.com/index.php?id=1100000004586986394&open=prime<br>http://factordb.com/index.php?id=1100000000808118270&open=prime<br>http://factordb.com/index.php?id=1100000004586986664&open=prime<br>http://factordb.com/index.php?id=1100000004586986798&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118215&base=25<br>http://factordb.com/index.php?showid=1100000003983674902&base=25<br>http://factordb.com/index.php?showid=1100000004909706420&base=25<br>http://factordb.com/index.php?showid=1100000004909733266&base=25<br>http://factordb.com/index.php?showid=1100000004909750102&base=25<br>http://factordb.com/index.php?showid=1100000004909770736&base=25<br>http://factordb.com/index.php?showid=1100000004586986394&base=25<br>http://factordb.com/index.php?showid=1100000000808118270&base=25<br>http://factordb.com/index.php?showid=1100000004586986664&base=25<br>http://factordb.com/index.php?showid=1100000004586986798&base=25<nowiki/>||85||100000|| |- ||26||25256~25259||85M₁₉₇₀₆₀B<br>M0₆₁₁₈₆2BB<br>J0₄₄₃₀₃KCB<br>6K₂₃₃₀₀5<br>LD0₂₀₉₇₅7<br>7₂₀₂₇₉OL<br>5₁₉₃₉₁6F<br>9GDK₁₅₉₂₀P<br>M₈₇₇₂P<br>K0₄₃₆₄I5||197063<br>61190<br>44307<br>23302<br>20978<br>20281<br>19393<br>15924<br>8773<br>4367||278839<br>86583<br>62694<br>32972<br>29684<br>28697<br>27440<br>22532<br>12414<br>6180||(5347×26¹⁹⁷⁰⁶¹−297)/25<br>22×26⁶¹¹⁸⁹+1649<br>19×26⁴⁴³⁰⁶+13843<br>(34×26²³³⁰¹−79)/5<br>559×26²⁰⁹⁷⁶+7<br>(7×26²⁰²⁸¹+11393)/25<br>(26¹⁹³⁹³+179)/5<br>(32569×26¹⁵⁹²¹+21)/5<br>(22×26⁸⁷⁷³+53)/25<br>20×26⁴³⁶⁶+473||http://factordb.com/index.php?id=1100000008573990023&open=prime<br>http://factordb.com/index.php?id=1100000003968169875&open=prime<br>http://factordb.com/index.php?id=1100000003968156595&open=prime<br>http://factordb.com/index.php?id=1100000003892628745&open=prime<br>http://factordb.com/index.php?id=1100000003892628658&open=prime<br>http://factordb.com/index.php?id=1100000003892628605&open=prime<br>http://factordb.com/index.php?id=1100000003850151202&open=prime<br>http://factordb.com/index.php?id=1100000003850155316&open=prime<br>http://factordb.com/index.php?id=1100000000758011195&open=prime<br>http://factordb.com/index.php?id=1100000002634136508&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008573990023&base=26<br>http://factordb.com/index.php?showid=1100000003968169875&base=26<br>http://factordb.com/index.php?showid=1100000003968156595&base=26<br>http://factordb.com/index.php?showid=1100000003892628745&base=26<br>http://factordb.com/index.php?showid=1100000003892628658&base=26<br>http://factordb.com/index.php?showid=1100000003892628605&base=26<br>http://factordb.com/index.php?showid=1100000003850151202&base=26<br>http://factordb.com/index.php?showid=1100000003850155316&base=26<br>http://factordb.com/index.php?showid=1100000000758011195&base=26<br>http://factordb.com/index.php?showid=1100000002634136508&base=26<nowiki/>||3||200000|| |- ||27||102852~102896||CA0F₈₈₈₈₃A<br>GNN0₇₈₇₉₅N<br>O44L₆₆₀₁₆7<br>NJ0₆₄₃₆₉H<br>ME₄₉₆₄₀9G<br>PH0₄₇₈₉₀1<br>QF₄₇₁₆₅AF5<br>J0₄₀₇₉₁PD<br>510₃₉₁₆₄I07<br>NGN0₃₆₃₂₉N||88887<br>78799<br>66020<br>64372<br>49643<br>47893<br>47169<br>40794<br>39169<br>36333||127230<br>112790<br>94499<br>92140<br>71058<br>68553<br>67516<br>58391<br>56065<br>52006||(234483×27⁸⁸⁸⁸⁴−145)/26<br>12308×27⁷⁸⁷⁹⁶+23<br>(457829×27⁶⁶⁰¹⁷−385)/26<br>640×27⁶⁴³⁷⁰+17<br>(293×27⁴⁹⁶⁴²−1736)/13<br>692×27⁴⁷⁸⁹¹+1<br>(691×27⁴⁷¹⁶⁸−95045)/26<br>19×27⁴⁰⁷⁹³+688<br>136×27³⁹¹⁶⁷+13129<br>17222×27³⁶³³⁰+23||http://factordb.com/index.php?id=1100000000808118233&open=prime<br>http://factordb.com/index.php?id=1100000004681348398&open=prime<br>http://factordb.com/index.php?id=1100000004374140861&open=prime<br>http://factordb.com/index.php?id=1100000004374138999&open=prime<br>http://factordb.com/index.php?id=1100000000819229859&open=prime<br>http://factordb.com/index.php?id=1100000004102754118&open=prime<br>http://factordb.com/index.php?id=1100000004102755880&open=prime<br>http://factordb.com/index.php?id=1100000004102758254&open=prime<br>http://factordb.com/index.php?id=1100000004102875088&open=prime<br>http://factordb.com/index.php?id=1100000004103372866&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118233&base=27<br>http://factordb.com/index.php?showid=1100000004681348398&base=27<br>http://factordb.com/index.php?showid=1100000004374140861&base=27<br>http://factordb.com/index.php?showid=1100000004374138999&base=27<br>http://factordb.com/index.php?showid=1100000000819229859&base=27<br>http://factordb.com/index.php?showid=1100000004102754118&base=27<br>http://factordb.com/index.php?showid=1100000004102755880&base=27<br>http://factordb.com/index.php?showid=1100000004102758254&base=27<br>http://factordb.com/index.php?showid=1100000004102875088&base=27<br>http://factordb.com/index.php?showid=1100000004103372866&base=27<nowiki/>||44||100000|| |- ||28||25528~25529||O4O₉₄₅₃₅9<br>5OA₃₁₂₃₈F<br>N6₂₄₀₅₁LR<br>D0₅₂₆₇77D<br>QO₄₂₃₉69<br>5₃₇₄₆8P<br>G0₁₈₉₉AN<br>A₁₄₂₃6F<br>5I₁₃₇₀F<br>5₁₃₃₂P8P||94538<br>31241<br>24054<br>5271<br>4242<br>3748<br>1902<br>1425<br>1372<br>1335||136812<br>45210<br>34810<br>7628<br>6139<br>5424<br>2753<br>2062<br>1985<br>1932||(6092×28⁹⁴⁵³⁶−143)/9<br>(4438×28³¹²³⁹+125)/27<br>(209×28²⁴⁰⁵³+3967)/9<br>13×28⁵²⁷⁰+5697<br>(242×28⁴²⁴¹−4679)/9<br>(5×28³⁷⁴⁸+2803)/27<br>16×28¹⁹⁰¹+303<br>(10×28¹⁴²⁵−2899)/27<br>(17×28¹³⁷¹−11)/3<br>(5×28¹³³⁵+426163)/27||http://factordb.com/index.php?id=1100000000808118231&open=prime<br>http://factordb.com/index.php?id=1100000003880455200&open=prime<br>http://factordb.com/index.php?id=1100000003879667576&open=prime<br>http://factordb.com/index.php?id=1100000003850151420&open=prime<br>http://factordb.com/index.php?id=1100000000840839934&open=prime<br>http://factordb.com/index.php?id=1100000003850161974&open=prime<br>http://factordb.com/index.php?id=1100000003850161973&open=prime<br>http://factordb.com/index.php?id=1100000000840839947&open=prime<br>http://factordb.com/index.php?id=1100000003850161972&open=prime<br>http://factordb.com/index.php?id=1100000003850161965&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118231&base=28<br>http://factordb.com/index.php?showid=1100000003880455200&base=28<br>http://factordb.com/index.php?showid=1100000003879667576&base=28<br>http://factordb.com/index.php?showid=1100000003850151420&base=28<br>http://factordb.com/index.php?showid=1100000000840839934&base=28<br>http://factordb.com/index.php?showid=1100000003850161974&base=28<br>http://factordb.com/index.php?showid=1100000003850161973&base=28<br>http://factordb.com/index.php?showid=1100000000840839947&base=28<br>http://factordb.com/index.php?showid=1100000003850161972&base=28<br>http://factordb.com/index.php?showid=1100000003850161965&base=28<nowiki/>||1||900000|| |- ||29||355242~355367||830₉₉₃₇₇4<br>GP5J₉₄₉₃₅<br>P05J₉₀₂₈₉<br>BBD0₈₈₈₈₈PB<br>8B₈₅₃₃₃G<br>L0₈₁₅₇₁5955<br>E0₇₇₃₇₂L7B<br>LPC₇₅₁₅₁9<br>JR0₇₄₆₂₂7<br>B₇₄₅₀₁0RP||99380<br>94938<br>90292<br>88893<br>85335<br>81576<br>77376<br>75154<br>74625<br>74504||145333<br>138837<br>132043<br>129997<br>124794<br>119297<br>113155<br>109905<br>109132<br>108955||235×29⁹⁹³⁷⁸+4<br>(397227×29⁹⁴⁹³⁵−19)/28<br>(588859×29⁹⁰²⁸⁹−19)/28<br>9583×29⁸⁸⁸⁹⁰+736<br>(235×29⁸⁵³³⁴+129)/14<br>21×29⁸¹⁵⁷⁵+129664<br>14×29⁷⁷³⁷⁵+17875<br>(4441×29⁷⁵¹⁵²−24)/7<br>578×29⁷⁴⁶²³+7<br>(11×29⁷⁴⁵⁰⁴−245655)/28||http://factordb.com/index.php?id=1100000008253882372&open=prime<br>http://factordb.com/index.php?id=1100000008253893542&open=prime<br>http://factordb.com/index.php?id=1100000008253899083&open=prime<br>http://factordb.com/index.php?id=1100000008253909183&open=prime<br>http://factordb.com/index.php?id=1100000008253921388&open=prime<br>http://factordb.com/index.php?id=1100000008253925955&open=prime<br>http://factordb.com/index.php?id=1100000008253931446&open=prime<br>http://factordb.com/index.php?id=1100000000808118236&open=prime<br>http://factordb.com/index.php?id=1100000008253934219&open=prime<br>http://factordb.com/index.php?id=1100000008253936120&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008253882372&base=29<br>http://factordb.com/index.php?showid=1100000008253893542&base=29<br>http://factordb.com/index.php?showid=1100000008253899083&base=29<br>http://factordb.com/index.php?showid=1100000008253909183&base=29<br>http://factordb.com/index.php?showid=1100000008253921388&base=29<br>http://factordb.com/index.php?showid=1100000008253925955&base=29<br>http://factordb.com/index.php?showid=1100000008253931446&base=29<br>http://factordb.com/index.php?showid=1100000000808118236&base=29<br>http://factordb.com/index.php?showid=1100000008253934219&base=29<br>http://factordb.com/index.php?showid=1100000008253936120&base=29<nowiki/>||125||100000|| |- ||30||2619||OT₃₄₂₀₅<br>I0₂₄₆₀₈D<br>5₄₈₈₂J<br>C0₁₀₂₂1<br>M0₅₄₇SS7<br>M₂₄₁QB<br>AN₂₀₆<br>50₁₆₄B<br>J₁₅₃QJ<br>J₉₄QQJ||34206<br>24610<br>4883<br>1024<br>551<br>243<br>207<br>166<br>155<br>97||50527<br>36352<br>7213<br>1513<br>814<br>359<br>306<br>245<br>229<br>144||25×30³⁴²⁰⁵−1<br>18×30²⁴⁶⁰⁹+13<br>(5×30⁴⁸⁸³+401)/29<br>12×30¹⁰²³+1<br>22×30⁵⁵⁰+26047<br>(22×30²⁴³+3139)/29<br>(313×30²⁰⁶−23)/29<br>5×30¹⁶⁵+11<br>(19×30¹⁵⁵+6071)/29<br>(19×30⁹⁷+188771)/29||http://factordb.com/index.php?id=1100000000800812865&open=prime<br>http://factordb.com/index.php?id=1100000003593967511&open=prime<br>http://factordb.com/index.php?id=1100000002327649423&open=prime<br>http://factordb.com/index.php?id=1100000000785448736&open=prime<br>http://factordb.com/index.php?id=1100000003593407988&open=prime<br>http://factordb.com/index.php?id=1100000003593408295&open=prime<br>http://factordb.com/index.php?id=1100000002327651073&open=prime<br>http://factordb.com/index.php?id=1100000002356282476&open=ecm<br>http://factordb.com/index.php?id=1100000003593409109&open=ecm<br>http://factordb.com/index.php?id=1100000003593409165&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000800812865&base=30<br>http://factordb.com/index.php?showid=1100000003593967511&base=30<br>http://factordb.com/index.php?showid=1100000002327649423&base=30<br>http://factordb.com/index.php?showid=1100000000785448736&base=30<br>http://factordb.com/index.php?showid=1100000003593407988&base=30<br>http://factordb.com/index.php?showid=1100000003593408295&base=30<br>http://factordb.com/index.php?showid=1100000002327651073&base=30<br>http://factordb.com/index.php?showid=1100000002356282476&base=30<br>http://factordb.com/index.php?showid=1100000003593409109&base=30<br>http://factordb.com/index.php?showid=1100000003593409165&base=30<nowiki/>||0||–|| |- ||31||569323~569400||2IIF₉₁₈₀₅<br>B0₈₈₃₀₉APO9<br>J0T₇₇₅₁₆<br>J090₇₇₁₂₈NNN<br>D₆₉₈₆₁QO<br>9MH0₆₈₆₃₇D<br>J₆₇₁₆₂D<br>N0₆₆₉₇₁32P<br>DDDQ0₆₄₀₈₈TD<br>U₆₃₈₆₁CM3||91808<br>88314<br>77518<br>77134<br>69863<br>68641<br>67163<br>66975<br>64094<br>63864||136918<br>131708<br>115608<br>115035<br>104191<br>102369<br>100165<br>99884<br>95587<br>95245||(4997×31⁹¹⁸⁰⁵−1)/2<br>11×31⁸⁸³¹³+322688<br>(17699×31⁷⁷⁵¹⁶−29)/30<br>18268×31⁷⁷¹³¹+22839<br>(13×31⁶⁹⁸⁶³+12407)/30<br>9348×31⁶⁸⁶³⁸+13<br>(19×31⁶⁷¹⁶³−199)/30<br>23×31⁶⁶⁹⁷⁴+2970<br>400205×31⁶⁴⁰⁹⁰+912<br>31⁶³⁸⁶⁴−17574||http://factordb.com/index.php?id=1100000007050395732&open=prime<br>http://factordb.com/index.php?id=1100000007050397309&open=prime<br>http://factordb.com/index.php?id=1100000007050398940&open=prime<br>http://factordb.com/index.php?id=1100000007050400178&open=prime<br>http://factordb.com/index.php?id=1100000006965878559&open=prime<br>http://factordb.com/index.php?id=1100000006965875678&open=prime<br>http://factordb.com/index.php?id=1100000006965873668&open=prime<br>http://factordb.com/index.php?id=1100000006965870538&open=prime<br>http://factordb.com/index.php?id=1100000006965868103&open=prime<br>http://factordb.com/index.php?id=1100000006965865343&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000007050395732&base=31<br>http://factordb.com/index.php?showid=1100000007050397309&base=31<br>http://factordb.com/index.php?showid=1100000007050398940&base=31<br>http://factordb.com/index.php?showid=1100000007050400178&base=31<br>http://factordb.com/index.php?showid=1100000006965878559&base=31<br>http://factordb.com/index.php?showid=1100000006965875678&base=31<br>http://factordb.com/index.php?showid=1100000006965873668&base=31<br>http://factordb.com/index.php?showid=1100000006965870538&base=31<br>http://factordb.com/index.php?showid=1100000006965868103&base=31<br>http://factordb.com/index.php?showid=1100000006965865343&base=31<nowiki/>||77||100000|| |- ||32||168882~169002||V₉₉₅₈₃63<br>6₈₉₀₇₄AF<br>8₇₇₇₀₀H<br>Q₇₇₄₀₁EQQQ3<br>8₇₇₂₄₉3<br>JM₇₆₀₂₈L<br>E₇₂₉₁₉IL<br>B0₆₇₆₈₀CB<br>GK₆₆₀₇₆F<br>KN₆₅₀₂₂||99585<br>89076<br>77701<br>77406<br>77250<br>76030<br>72921<br>67683<br>66078<br>65023||149891<br>134073<br>116952<br>116508<br>116273<br>114437<br>109757<br>101873<br>99458<br>97870||32⁹⁹⁵⁸⁵−829<br>(6×32⁸⁹⁰⁷⁶+4241)/31<br>(8×32⁷⁷⁷⁰¹+271)/31<br>(26×32⁷⁷⁴⁰⁶−390071011)/31<br>(8×32⁷⁷²⁵⁰−163)/31<br>(611×32⁷⁶⁰²⁹−53)/31<br>(14×32⁷²⁹²¹+4171)/31<br>11×32⁶⁷⁶⁸²+395<br>(516×32⁶⁶⁰⁷⁷−175)/31<br>(643×32⁶⁵⁰²²−23)/31||http://factordb.com/index.php?id=1100000005514892191&open=prime<br>http://factordb.com/index.php?id=1100000005514897129&open=prime<br>http://factordb.com/index.php?id=1100000005514901700&open=prime<br>http://factordb.com/index.php?id=1100000005514915338&open=prime<br>http://factordb.com/index.php?id=1100000005514918574&open=prime<br>http://factordb.com/index.php?id=1100000005514922523&open=prime<br>http://factordb.com/index.php?id=1100000004591654373&open=prime<br>http://factordb.com/index.php?id=1100000004591654467&open=prime<br>http://factordb.com/index.php?id=1100000004591654632&open=prime<br>http://factordb.com/index.php?id=1100000004591654952&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005514892191&base=32<br>http://factordb.com/index.php?showid=1100000005514897129&base=32<br>http://factordb.com/index.php?showid=1100000005514901700&base=32<br>http://factordb.com/index.php?showid=1100000005514915338&base=32<br>http://factordb.com/index.php?showid=1100000005514918574&base=32<br>http://factordb.com/index.php?showid=1100000005514922523&base=32<br>http://factordb.com/index.php?showid=1100000004591654373&base=32<br>http://factordb.com/index.php?showid=1100000004591654467&base=32<br>http://factordb.com/index.php?showid=1100000004591654632&base=32<br>http://factordb.com/index.php?showid=1100000004591654952&base=32<nowiki/>||120||100000|| |- ||33||280012~280093||DP₉₅₀₉₃M5<br>HJ0₉₄₂₉₅J<br>90₉₃₅₉₇Q<br>9F0₉₃₁₅₇N<br>7₈₉₄₄₉333H<br>K3₈₀₇₅₁6K<br>D₈₀₁₀₇9UD<br>VFU₇₂₂₀₄FK<br>J₆₈₇₁₅2BJ<br>DF0₆₈₃₆₇J||95096<br>94298<br>93599<br>93160<br>89453<br>80754<br>80110<br>72208<br>68718<br>68370||144405<br>143193<br>142131<br>141465<br>135835<br>122626<br>121648<br>109649<br>104350<br>103821||(441×33⁹⁵⁰⁹⁵−3833)/32<br>580×33⁹⁴²⁹⁶+19<br>9×33⁹³⁵⁹⁸+26<br>312×33⁹³¹⁵⁸+23<br>(7×33⁸⁹⁴⁵³−4743239)/32<br>(643×33⁸⁰⁷⁵³+3709)/32<br>(13×33⁸⁰¹¹⁰−121453)/32<br>(16623×33⁷²²⁰⁶−8095)/16<br>(19×33⁶⁸⁷¹⁸−600883)/32<br>444×33⁶⁸³⁶⁸+19||http://factordb.com/index.php?id=1100000005652348775&open=prime<br>http://factordb.com/index.php?id=1100000005652362811&open=prime<br>http://factordb.com/index.php?id=1100000005652375073&open=prime<br>http://factordb.com/index.php?id=1100000005652389776&open=prime<br>http://factordb.com/index.php?id=1100000005652430746&open=prime<br>http://factordb.com/index.php?id=1100000005652446200&open=prime<br>http://factordb.com/index.php?id=1100000005652461592&open=prime<br>http://factordb.com/index.php?id=1100000004614764298&open=prime<br>http://factordb.com/index.php?id=1100000004614770536&open=prime<br>http://factordb.com/index.php?id=1100000004614784274&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005652348775&base=33<br>http://factordb.com/index.php?showid=1100000005652362811&base=33<br>http://factordb.com/index.php?showid=1100000005652375073&base=33<br>http://factordb.com/index.php?showid=1100000005652389776&base=33<br>http://factordb.com/index.php?showid=1100000005652430746&base=33<br>http://factordb.com/index.php?showid=1100000005652446200&base=33<br>http://factordb.com/index.php?showid=1100000005652461592&base=33<br>http://factordb.com/index.php?showid=1100000004614764298&base=33<br>http://factordb.com/index.php?showid=1100000004614770536&base=33<br>http://factordb.com/index.php?showid=1100000004614784274&base=33<nowiki/>||81||100000|| |- ||34||184785~184832||GFGC₉₉₉₉₆5<br>90₉₇₉₅₀FJ<br>NM0₈₅₂₁₈KX<br>F₈₃₁₈₉H2HP<br>P₇₉₄₄₁444P<br>6₇₇₀₂₇8X<br>XQIQ₇₂₂₄₁D<br>T₆₆₅₃₀IF<br>4₆₆₁₅₂B<br>2EEC₆₆₀₃₉7||100000<br>97953<br>85222<br>83193<br>79445<br>77029<br>72245<br>66532<br>66153<br>66043||153148<br>150013<br>130516<br>127408<br>121669<br>117968<br>110642<br>101893<br>101312<br>101143||(209246×34⁹⁹⁹⁹⁷−81)/11<br>9×34⁹⁷⁹⁵²+529<br>804×34⁸⁵²²⁰+713<br>(5×34⁸³¹⁹³+700233)/11<br>(25×34⁷⁹⁴⁴⁵−28062367)/33<br>(2×34⁷⁷⁰²⁹+1043)/11<br>(1288676×34⁷²²⁴²−455)/33<br>(29×34⁶⁶⁵³²−12833)/33<br>(4×34⁶⁶¹⁵³+227)/33<br>(30826×34⁶⁶⁰⁴⁰−59)/11||http://factordb.com/index.php?id=1100000004702891268&open=prime<br>http://factordb.com/index.php?id=1100000004702894713&open=prime<br>http://factordb.com/index.php?id=1100000004702900996&open=prime<br>http://factordb.com/index.php?id=1100000004702910376&open=prime<br>http://factordb.com/index.php?id=1100000004702913746&open=prime<br>http://factordb.com/index.php?id=1100000004702918600&open=prime<br>http://factordb.com/index.php?id=1100000004399656529&open=prime<br>http://factordb.com/index.php?id=1100000004399657696&open=prime<br>http://factordb.com/index.php?id=1100000004399658651&open=prime<br>http://factordb.com/index.php?id=1100000004399659716&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004702891268&base=34<br>http://factordb.com/index.php?showid=1100000004702894713&base=34<br>http://factordb.com/index.php?showid=1100000004702900996&base=34<br>http://factordb.com/index.php?showid=1100000004702910376&base=34<br>http://factordb.com/index.php?showid=1100000004702913746&base=34<br>http://factordb.com/index.php?showid=1100000004702918600&base=34<br>http://factordb.com/index.php?showid=1100000004399656529&base=34<br>http://factordb.com/index.php?showid=1100000004399657696&base=34<br>http://factordb.com/index.php?showid=1100000004399658651&base=34<br>http://factordb.com/index.php?showid=1100000004399659716&base=34<nowiki/>||47||100000|| |- ||35||720002~720062||N0N₉₉₉₇₁9<br>V0₈₃₆₆₉E73<br>N₈₁₅₆₃K7N<br>BJ0₈₁₂₇₉N<br>J0₈₀₀₆₂FUH<br>43V₇₉₇₅₄<br>9₇₆₆₀₀K3<br>LB₇₁₃₆₆PB<br>Q₆₄₁₅₀H<br>50₆₃₃₉₇5R||99974<br>83673<br>81566<br>81282<br>80066<br>79756<br>76602<br>71369<br>64151<br>63400||154367<br>129197<br>125944<br>125505<br>123628<br>123148<br>118279<br>110199<br>99054<br>97894||(27393×35⁹⁹⁹⁷²−499)/34<br>31×35⁸³⁶⁷²+17398<br>(23×35⁸¹⁵⁶⁶−144013)/34<br>404×35⁸¹²⁸⁰+23<br>19×35⁸⁰⁰⁶⁵+19442<br>(4893×35⁷⁹⁷⁵⁴−31)/34<br>(9×35⁷⁶⁶⁰²+12877)/34<br>(725×35⁷¹³⁶⁸+16649)/34<br>(13×35⁶⁴¹⁵¹−166)/17<br>5×35⁶³³⁹⁹+202||http://factordb.com/index.php?id=1100000008248342445&open=prime<br>http://factordb.com/index.php?id=1100000008248353306&open=prime<br>http://factordb.com/index.php?id=1100000008248375642&open=prime<br>http://factordb.com/index.php?id=1100000008248397018&open=prime<br>http://factordb.com/index.php?id=1100000008248412468&open=prime<br>http://factordb.com/index.php?id=1100000008248418540&open=prime<br>http://factordb.com/index.php?id=1100000008248423670&open=prime<br>http://factordb.com/index.php?id=1100000008192119974&open=prime<br>http://factordb.com/index.php?id=1100000008192126630&open=prime<br>http://factordb.com/index.php?id=1100000008192129294&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008248342445&base=35<br>http://factordb.com/index.php?showid=1100000008248353306&base=35<br>http://factordb.com/index.php?showid=1100000008248375642&base=35<br>http://factordb.com/index.php?showid=1100000008248397018&base=35<br>http://factordb.com/index.php?showid=1100000008248412468&base=35<br>http://factordb.com/index.php?showid=1100000008248418540&base=35<br>http://factordb.com/index.php?showid=1100000008248423670&base=35<br>http://factordb.com/index.php?showid=1100000008192119974&base=35<br>http://factordb.com/index.php?showid=1100000008192126630&base=35<br>http://factordb.com/index.php?showid=1100000008192129294&base=35<nowiki/>||60||100000|| |- ||36||35286~35290||P₈₁₉₉₃SZ<br>S0₇₅₀₀₇8H<br>7K₂₆₅₆₇Z<br>J₁₀₁₁₇LJ<br>VL0₇₂₅₈J<br>EO0₆₁₇₇V<br>FZ₅₇₇₇3P<br>T09₄₆₁₈1<br>RY₄₅₆₂H<br>OZ₃₉₃₂AZ||81995<br>75010<br>26569<br>10119<br>7261<br>6180<br>5780<br>4621<br>4564<br>3935||127609<br>116739<br>41349<br>15748<br>11301<br>9618<br>8996<br>7192<br>7103<br>6124||(5×36⁸¹⁹⁹⁵+821)/7<br>28×36⁷⁵⁰⁰⁹+305<br>(53×36²⁶⁵⁶⁸+101)/7<br>(19×36¹⁰¹¹⁹+2501)/35<br>1137×36⁷²⁵⁹+19<br>528×36⁶¹⁷⁸+31<br>16×36⁵⁷⁷⁹−1163<br>(36549×36⁴⁶¹⁹−289)/35<br>(979×36⁴⁵⁶³−629)/35<br>25×36³⁹³⁴−901||http://factordb.com/index.php?id=1100000002394962083&open=prime<br>http://factordb.com/index.php?id=1100000004020085177&open=prime<br>http://factordb.com/index.php?id=1100000003896952461&open=prime<br>http://factordb.com/index.php?id=1100000003807362491&open=prime<br>http://factordb.com/index.php?id=1100000003807362489&open=prime<br>http://factordb.com/index.php?id=1100000003807362488&open=prime<br>http://factordb.com/index.php?id=1100000003807362487&open=prime<br>http://factordb.com/index.php?id=1100000003807362486&open=prime<br>http://factordb.com/index.php?id=1100000003807362485&open=prime<br>http://factordb.com/index.php?id=1100000000840634476&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000002394962083&base=36<br>http://factordb.com/index.php?showid=1100000004020085177&base=36<br>http://factordb.com/index.php?showid=1100000003896952461&base=36<br>http://factordb.com/index.php?showid=1100000003807362491&base=36<br>http://factordb.com/index.php?showid=1100000003807362489&base=36<br>http://factordb.com/index.php?showid=1100000003807362488&base=36<br>http://factordb.com/index.php?showid=1100000003807362487&base=36<br>http://factordb.com/index.php?showid=1100000003807362486&base=36<br>http://factordb.com/index.php?showid=1100000003807362485&base=36<br>http://factordb.com/index.php?showid=1100000000840634476&base=36<nowiki/>||4||200000|| |} == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] huqv2blo2nvee4rjfhytrsdvi7j53bp 2819703 2819701 2026-07-28T19:21:14Z Athene241 3100061 /* Solve the problem */ 2819703 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484; or for prime bases ''b'': https://oeis.org/A122396), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored). The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]]. To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''^''n''+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1 we only used it to sieve the sequence ''a''×''b''^''n''+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''^''n''+''c'', but 2 may not divide (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the "easy" bases (bases ''b'' with ≤ 150 primes > 10²⁹⁹ (base ''b'' = 26 has 83 known primes > 10²⁹⁹ and 3 unsolved families, base ''b'' = 36 has 75 known primes > 10²⁹⁹ and 4 unsolved families, base ''b'' = 17 has 99 known primes > 10²⁹⁹ and 18 unsolved families, base ''b'' = 21 has 80 known primes > 10²⁹⁹ and 12 unsolved families, base ''b'' = 19 has 201 known primes > 10²⁹⁹ and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation. == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == Condensed table for bases 2 ≤ ''b'' ≤ 36 == {|class="wikitable" ||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write "''d''_''n''" if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)|| |- ||2||1||11||2||1||3||http://factordb.com/index.php?id=3&open=ecm||http://factordb.com/index.php?showid=3&base=2||0||–|| |- ||3||3||111<br>21<br>12||3<br>2<br>2||2<br>1<br>1||13<br>7<br>5||http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=13&base=3<br>http://factordb.com/index.php?showid=7&base=3<br>http://factordb.com/index.php?showid=5&base=3<nowiki/>||0||–|| |- ||4||5||221<br>31<br>23<br>13<br>11||3<br>2<br>2<br>2<br>2||2<br>2<br>2<br>1<br>1||41<br>13<br>11<br>7<br>5||http://factordb.com/index.php?id=41&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=41&base=4<br>http://factordb.com/index.php?showid=13&base=4<br>http://factordb.com/index.php?showid=11&base=4<br>http://factordb.com/index.php?showid=7&base=4<br>http://factordb.com/index.php?showid=5&base=4<nowiki/>||0||–|| |- ||5||22||10₉₃13<br>300031<br>44441<br>33331<br>33001<br>30301<br>14444<br>10103<br>3101<br>414||96<br>6<br>5<br>5<br>5<br>5<br>5<br>5<br>4<br>3||67<br>4<br>4<br>4<br>4<br>4<br>4<br>3<br>3<br>3||5⁹⁵+8<br>9391<br>3121<br>2341<br>2251<br>1951<br>1249<br>653<br>401<br>109||http://factordb.com/index.php?id=1100000000034686071&open=ecm<br>http://factordb.com/index.php?id=9391&open=ecm<br>http://factordb.com/index.php?id=3121&open=ecm<br>http://factordb.com/index.php?id=2341&open=ecm<br>http://factordb.com/index.php?id=2251&open=ecm<br>http://factordb.com/index.php?id=1951&open=ecm<br>http://factordb.com/index.php?id=1249&open=ecm<br>http://factordb.com/index.php?id=653&open=ecm<br>http://factordb.com/index.php?id=401&open=ecm<br>http://factordb.com/index.php?id=109&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000034686071&base=5<br>http://factordb.com/index.php?showid=9391&base=5<br>http://factordb.com/index.php?showid=3121&base=5<br>http://factordb.com/index.php?showid=2341&base=5<br>http://factordb.com/index.php?showid=2251&base=5<br>http://factordb.com/index.php?showid=1951&base=5<br>http://factordb.com/index.php?showid=1249&base=5<br>http://factordb.com/index.php?showid=653&base=5<br>http://factordb.com/index.php?showid=401&base=5<br>http://factordb.com/index.php?showid=109&base=5<nowiki/>||0||–|| |- ||6||11||40041<br>4441<br>4401<br>51<br>45<br>35<br>31<br>25<br>21<br>15||5<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||4<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||5209<br>1033<br>1009<br>31<br>29<br>23<br>19<br>17<br>13<br>11||http://factordb.com/index.php?id=5209&open=ecm<br>http://factordb.com/index.php?id=1033&open=ecm<br>http://factordb.com/index.php?id=1009&open=ecm<br>http://factordb.com/index.php?id=31&open=ecm<br>http://factordb.com/index.php?id=29&open=ecm<br>http://factordb.com/index.php?id=23&open=ecm<br>http://factordb.com/index.php?id=19&open=ecm<br>http://factordb.com/index.php?id=17&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<nowiki/>||http://factordb.com/index.php?showid=5209&base=6<br>http://factordb.com/index.php?showid=1033&base=6<br>http://factordb.com/index.php?showid=1009&base=6<br>http://factordb.com/index.php?showid=31&base=6<br>http://factordb.com/index.php?showid=29&base=6<br>http://factordb.com/index.php?showid=23&base=6<br>http://factordb.com/index.php?showid=19&base=6<br>http://factordb.com/index.php?showid=17&base=6<br>http://factordb.com/index.php?showid=13&base=6<br>http://factordb.com/index.php?showid=11&base=6<nowiki/>||0||–|| |- ||7||71||3₁₆1<br>510₇1<br>3₆01<br>1100021<br>531101<br>351101<br>300053<br>150001<br>100121<br>40054||17<br>10<br>8<br>7<br>6<br>6<br>6<br>6<br>6<br>5||15<br>9<br>7<br>6<br>5<br>5<br>5<br>5<br>5<br>4||(7¹⁷−5)/2<br>36×7⁸+1<br>(7⁸−47)/2<br>134471<br>91631<br>62819<br>50459<br>28813<br>16871<br>9643||http://factordb.com/index.php?id=116315256993601&open=ecm<br>http://factordb.com/index.php?id=207532837&open=ecm<br>http://factordb.com/index.php?id=2882377&open=ecm<br>http://factordb.com/index.php?id=134471&open=ecm<br>http://factordb.com/index.php?id=91631&open=ecm<br>http://factordb.com/index.php?id=62819&open=ecm<br>http://factordb.com/index.php?id=50459&open=ecm<br>http://factordb.com/index.php?id=28813&open=ecm<br>http://factordb.com/index.php?id=16871&open=ecm<br>http://factordb.com/index.php?id=9643&open=ecm<nowiki/>||http://factordb.com/index.php?showid=116315256993601&base=7<br>http://factordb.com/index.php?showid=207532837&base=7<br>http://factordb.com/index.php?showid=2882377&base=7<br>http://factordb.com/index.php?showid=134471&base=7<br>http://factordb.com/index.php?showid=91631&base=7<br>http://factordb.com/index.php?showid=62819&base=7<br>http://factordb.com/index.php?showid=50459&base=7<br>http://factordb.com/index.php?showid=28813&base=7<br>http://factordb.com/index.php?showid=16871&base=7<br>http://factordb.com/index.php?showid=9643&base=7<nowiki/>||0||–|| |- ||8||75||4₂₂₀7<br>5₁₃25<br>7₁₂1<br>77774₆1<br>74₇1<br>4₈1<br>5₅025<br>5550525<br>5500525<br>4₅77||221<br>15<br>13<br>11<br>9<br>9<br>8<br>7<br>7<br>7||200<br>14<br>12<br>10<br>9<br>8<br>8<br>7<br>7<br>7||(4×8²²¹+17)/7<br>(5×8¹⁵−173)/7<br>8¹³−7<br>(28669×8⁷−25)/7<br>(53×8⁸−25)/7<br>(4×8⁹−25)/7<br>(5×8⁸−2413)/7<br>1495381<br>1474901<br>(4×8⁷+185)/7||http://factordb.com/index.php?id=1100000000416605822&open=ecm<br>http://factordb.com/index.php?id=25131694349141&open=ecm<br>http://factordb.com/index.php?id=549755813881&open=ecm<br>http://factordb.com/index.php?id=8589035809&open=ecm<br>http://factordb.com/index.php?id=127027489&open=ecm<br>http://factordb.com/index.php?id=76695841&open=ecm<br>http://factordb.com/index.php?id=11983381&open=ecm<br>http://factordb.com/index.php?id=1495381&open=ecm<br>http://factordb.com/index.php?id=1474901&open=ecm<br>http://factordb.com/index.php?id=1198399&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000416605822&base=8<br>http://factordb.com/index.php?showid=25131694349141&base=8<br>http://factordb.com/index.php?showid=549755813881&base=8<br>http://factordb.com/index.php?showid=8589035809&base=8<br>http://factordb.com/index.php?showid=127027489&base=8<br>http://factordb.com/index.php?showid=76695841&base=8<br>http://factordb.com/index.php?showid=11983381&base=8<br>http://factordb.com/index.php?showid=1495381&base=8<br>http://factordb.com/index.php?showid=1474901&base=8<br>http://factordb.com/index.php?showid=1198399&base=8<nowiki/>||0||–|| |- ||9||151||30₁₁₅₈11<br>27₆₈₆07<br>76₃₂₉2<br>561₃₆<br>10₂₅57<br>30₂₀51<br>8₁₉335<br>727₁₅07<br>51₁₃61<br>10₁₁507||1161<br>689<br>331<br>38<br>28<br>23<br>22<br>19<br>16<br>15||1108<br>657<br>316<br>37<br>26<br>22<br>21<br>19<br>16<br>14||3×9¹¹⁶⁰+10<br>(23×9⁶⁸⁸−511)/8<br>(31×9³³⁰−19)/4<br>(409×9³⁶−1)/8<br>9²⁷+52<br>3×9²²+46<br>9²²−454<br>(527×9¹⁷−511)/8<br>(41×9¹⁵+359)/8<br>9¹⁴+412||http://factordb.com/index.php?id=1100000002376318423&open=prime<br>http://factordb.com/index.php?id=1100000002495467486&open=prime<br>http://factordb.com/index.php?id=1100000002359003642&open=prime<br>http://factordb.com/index.php?id=1100000001554010824&open=ecm<br>http://factordb.com/index.php?id=1100000002512830927&open=ecm<br>http://factordb.com/index.php?id=1100000000032261811&open=ecm<br>http://factordb.com/index.php?id=1100000002495736583&open=ecm<br>http://factordb.com/index.php?id=1100000003446800389&open=ecm<br>http://factordb.com/index.php?id=1055192051985121&open=ecm<br>http://factordb.com/index.php?id=22876792455373&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002376318423&base=9<br>http://factordb.com/index.php?showid=1100000002495467486&base=9<br>http://factordb.com/index.php?showid=1100000002359003642&base=9<br>http://factordb.com/index.php?showid=1100000001554010824&base=9<br>http://factordb.com/index.php?showid=1100000002512830927&base=9<br>http://factordb.com/index.php?showid=1100000000032261811&base=9<br>http://factordb.com/index.php?showid=1100000002495736583&base=9<br>http://factordb.com/index.php?showid=1100000003446800389&base=9<br>http://factordb.com/index.php?showid=1055192051985121&base=9<br>http://factordb.com/index.php?showid=22876792455373&base=9<nowiki/>||0||–|| |- ||10||77||50₂₈27<br>5₁₁1<br>805₅1<br>66600049<br>66000049<br>60₅49<br>220₅1<br>5200007<br>946669<br>666649||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||5×10³⁰+27<br>(5×10¹²−41)/9<br>(725×10⁶−41)/9<br>66600049<br>66000049<br>6×10⁷+49<br>22×10⁶+1<br>5200007<br>946669<br>666649||http://factordb.com/index.php?id=1100000000204142046&open=ecm<br>http://factordb.com/index.php?id=555555555551&open=ecm<br>http://factordb.com/index.php?id=80555551&open=ecm<br>http://factordb.com/index.php?id=66600049&open=ecm<br>http://factordb.com/index.php?id=66000049&open=ecm<br>http://factordb.com/index.php?id=60000049&open=ecm<br>http://factordb.com/index.php?id=22000001&open=ecm<br>http://factordb.com/index.php?id=5200007&open=ecm<br>http://factordb.com/index.php?id=946669&open=ecm<br>http://factordb.com/index.php?id=666649&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000204142046&base=10<br>http://factordb.com/index.php?showid=555555555551&base=10<br>http://factordb.com/index.php?showid=80555551&base=10<br>http://factordb.com/index.php?showid=66600049&base=10<br>http://factordb.com/index.php?showid=66000049&base=10<br>http://factordb.com/index.php?showid=60000049&base=10<br>http://factordb.com/index.php?showid=22000001&base=10<br>http://factordb.com/index.php?showid=5200007&base=10<br>http://factordb.com/index.php?showid=946669&base=10<br>http://factordb.com/index.php?showid=666649&base=10<nowiki/>||0||–|| |- ||11||1068||57₆₂₆₆₈<br>557₁₀₁₁<br>7₇₅₉44<br>A₇₁₃58<br>85₂₂₀05<br>507₂₀₆<br>5₁₆₁2A<br>50₁₂₆57<br>10₁₂₅51<br>326₁₂₂||62669<br>1013<br>761<br>715<br>223<br>208<br>163<br>129<br>128<br>124||65263<br>1055<br>793<br>745<br>233<br>217<br>170<br>134<br>133<br>129||(57×11⁶²⁶⁶⁸−7)/10<br>(607×11¹⁰¹¹−7)/10<br>(7×11⁷⁶¹−367)/10<br>11⁷¹⁵−58<br>(17×11²²²−111)/2<br>(557×11²⁰⁶−7)/10<br>(11¹⁶³−57)/2<br>5×11¹²⁸+62<br>11¹²⁷+56<br>(178×11¹²²−3)/5||http://factordb.com/index.php?id=1100000003573679860&open=prime<br>http://factordb.com/index.php?id=1100000002361376522&open=prime<br>http://factordb.com/index.php?id=1100000002505568840&open=prime<br>http://factordb.com/index.php?id=1100000003576826487&open=prime<br>http://factordb.com/index.php?id=1100000003576826769&open=ecm<br>http://factordb.com/index.php?id=1100000002518512744&open=ecm<br>http://factordb.com/index.php?id=1100000002391585327&open=ecm<br>http://factordb.com/index.php?id=1100000002632393378&open=ecm<br>http://factordb.com/index.php?id=1100000002391531300&open=ecm<br>http://factordb.com/index.php?id=1100000003576826781&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003573679860&base=11<br>http://factordb.com/index.php?showid=1100000002361376522&base=11<br>http://factordb.com/index.php?showid=1100000002505568840&base=11<br>http://factordb.com/index.php?showid=1100000003576826487&base=11<br>http://factordb.com/index.php?showid=1100000003576826769&base=11<br>http://factordb.com/index.php?showid=1100000002518512744&base=11<br>http://factordb.com/index.php?showid=1100000002391585327&base=11<br>http://factordb.com/index.php?showid=1100000002632393378&base=11<br>http://factordb.com/index.php?showid=1100000002391531300&base=11<br>http://factordb.com/index.php?showid=1100000003576826781&base=11<nowiki/>||0||–|| |- ||12||106||40₃₉77<br>B0₂₇9B<br>B₆99B<br>AA0₅1<br>B00099B<br>AAA0001<br>BBBAA1<br>A00065<br>44AAA1<br>BBBB1||42<br>30<br>9<br>8<br>7<br>7<br>6<br>6<br>6<br>5||45<br>33<br>10<br>9<br>8<br>8<br>7<br>7<br>7<br>6||4×12⁴¹+91<br>11×12²⁹+119<br>12⁹−313<br>130×12⁶+1<br>32847239<br>32555521<br>2985817<br>2488397<br>1097113<br>248821||http://factordb.com/index.php?id=1100000002375054575&open=ecm<br>http://factordb.com/index.php?id=1100000002354113100&open=ecm<br>http://factordb.com/index.php?id=5159780039&open=ecm<br>http://factordb.com/index.php?id=388177921&open=ecm<br>http://factordb.com/index.php?id=32847239&open=ecm<br>http://factordb.com/index.php?id=32555521&open=ecm<br>http://factordb.com/index.php?id=2985817&open=ecm<br>http://factordb.com/index.php?id=2488397&open=ecm<br>http://factordb.com/index.php?id=1097113&open=ecm<br>http://factordb.com/index.php?id=248821&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002375054575&base=12<br>http://factordb.com/index.php?showid=1100000002354113100&base=12<br>http://factordb.com/index.php?showid=5159780039&base=12<br>http://factordb.com/index.php?showid=388177921&base=12<br>http://factordb.com/index.php?showid=32847239&base=12<br>http://factordb.com/index.php?showid=32555521&base=12<br>http://factordb.com/index.php?showid=2985817&base=12<br>http://factordb.com/index.php?showid=2488397&base=12<br>http://factordb.com/index.php?showid=1097113&base=12<br>http://factordb.com/index.php?showid=248821&base=12<nowiki/>||0||–|| |- ||13||3197||A3₅₉₂₁₉₇A<br>95₁₉₇₄₂₀<br>80₃₂₀₁₇111<br>C5₂₃₇₅₅C<br>C₁₀₆₃₁92<br>B0₆₅₄₀BBA<br>390₆₂₆₆1<br>1770₂₇₀₃17<br>720₂₂₉₇2<br>930₁₅₅₁1||592199<br>197421<br>32021<br>23757<br>10633<br>6544<br>6269<br>2708<br>2300<br>1554||659677<br>219916<br>35670<br>26464<br>11845<br>7290<br>6983<br>3016<br>2562<br>1731||(41×13⁵⁹²¹⁹⁸+27)/4<br>(113×13¹⁹⁷⁴²⁰−5)/12<br>8×13³²⁰²⁰+183<br>(149×13²³⁷⁵⁶+79)/12<br>13¹⁰⁶³³−50<br>11×13⁶⁵⁴³+2012<br>48×13⁶²⁶⁷+1<br>267×13²⁷⁰⁵+20<br>93×13²²⁹⁸+2<br>120×13¹⁵⁵²+1||http://factordb.com/index.php?id=1100000005489162806&open=prime<br>http://factordb.com/index.php?id=1100000003943359311&open=prime<br>http://factordb.com/index.php?id=1100000000490878060&open=prime<br>http://factordb.com/index.php?id=1100000003590647776&open=prime<br>http://factordb.com/index.php?id=1100000003590493750&open=prime<br>http://factordb.com/index.php?id=1100000002616382906&open=prime<br>http://factordb.com/index.php?id=1100000000765961441&open=prime<br>http://factordb.com/index.php?id=1100000003590430825&open=prime<br>http://factordb.com/index.php?id=1100000002632396910&open=prime<br>http://factordb.com/index.php?id=1100000000765961452&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005489162806&base=13<br>http://factordb.com/index.php?showid=1100000003943359311&base=13<br>http://factordb.com/index.php?showid=1100000000490878060&base=13<br>http://factordb.com/index.php?showid=1100000003590647776&base=13<br>http://factordb.com/index.php?showid=1100000003590493750&base=13<br>http://factordb.com/index.php?showid=1100000002616382906&base=13<br>http://factordb.com/index.php?showid=1100000000765961441&base=13<br>http://factordb.com/index.php?showid=1100000003590430825&base=13<br>http://factordb.com/index.php?showid=1100000002632396910&base=13<br>http://factordb.com/index.php?showid=1100000000765961452&base=13<nowiki/>||0||–|| |- ||14||650||4D₁₉₆₉₈<br>34D₇₀₈<br>8D₁₄₁85<br>8₈₆B<br>40₈₃49<br>8C₇₉3<br>18₇₉B<br>6B₇₇2B<br>4₆₃09<br>A₅₉3||19699<br>710<br>144<br>87<br>86<br>81<br>81<br>80<br>65<br>60||22578<br>814<br>165<br>100<br>99<br>93<br>92<br>92<br>74<br>69||5×14¹⁹⁶⁹⁸−1<br>47×14⁷⁰⁸−1<br>9×14¹⁴³−79<br>(8×14⁸⁷+31)/13<br>4×14⁸⁵+65<br>(116×14⁸⁰−129)/13<br>(21×14⁸⁰+31)/13<br>(89×14⁷⁹−1649)/13<br>(4×14⁶⁵−667)/13<br>(10×14⁶⁰−101)/13||http://factordb.com/index.php?id=1100000000884560233&open=prime<br>http://factordb.com/index.php?id=1100000001540144903&open=prime<br>http://factordb.com/index.php?id=1100000003575856650&open=ecm<br>http://factordb.com/index.php?id=1100000002321014379&open=ecm<br>http://factordb.com/index.php?id=1100000000823937973&open=ecm<br>http://factordb.com/index.php?id=1100000002631073246&open=ecm<br>http://factordb.com/index.php?id=1100000002384401372&open=ecm<br>http://factordb.com/index.php?id=1100000002631077787&open=ecm<br>http://factordb.com/index.php?id=1100000000840126683&open=ecm<br>http://factordb.com/index.php?id=1100000002321038522&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000884560233&base=14<br>http://factordb.com/index.php?showid=1100000001540144903&base=14<br>http://factordb.com/index.php?showid=1100000003575856650&base=14<br>http://factordb.com/index.php?showid=1100000002321014379&base=14<br>http://factordb.com/index.php?showid=1100000000823937973&base=14<br>http://factordb.com/index.php?showid=1100000002631073246&base=14<br>http://factordb.com/index.php?showid=1100000002384401372&base=14<br>http://factordb.com/index.php?showid=1100000002631077787&base=14<br>http://factordb.com/index.php?showid=1100000000840126683&base=14<br>http://factordb.com/index.php?showid=1100000002321038522&base=14<nowiki/>||0||–|| |- ||15||1284||7₁₅₅97<br>E₁₄₅397<br>96₁₀₄08<br>7₇₃CE<br>7₅₉CCE<br>50₃₃17<br>EB₃₁<br>6330₂₆1<br>7050₂₄B<br>B70₂₄1||157<br>148<br>107<br>75<br>62<br>36<br>32<br>30<br>28<br>27||185<br>175<br>126<br>88<br>73<br>42<br>38<br>35<br>33<br>32||(15¹⁵⁷+59)/2<br>15¹⁴⁸−2558<br>(66×15¹⁰⁶−619)/7<br>(15⁷⁵+163)/2<br>(15⁶²+2413)/2<br>5×15³⁵+22<br>(207×15³¹−11)/14<br>1398×15²⁷+1<br>1580×15²⁵+11<br>172×15²⁵+1||http://factordb.com/index.php?id=1100000002454891840&open=ecm<br>http://factordb.com/index.php?id=1100000002454900849&open=ecm<br>http://factordb.com/index.php?id=1100000000823937997&open=ecm<br>http://factordb.com/index.php?id=1100000003588407143&open=ecm<br>http://factordb.com/index.php?id=1100000003588407386&open=ecm<br>http://factordb.com/index.php?id=1100000002632398579&open=ecm<br>http://factordb.com/index.php?id=1100000002321033312&open=ecm<br>http://factordb.com/index.php?id=1100000002391199877&open=ecm<br>http://factordb.com/index.php?id=1100000003588407806&open=ecm<br>http://factordb.com/index.php?id=1100000000851967288&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002454891840&base=15<br>http://factordb.com/index.php?showid=1100000002454900849&base=15<br>http://factordb.com/index.php?showid=1100000000823937997&base=15<br>http://factordb.com/index.php?showid=1100000003588407143&base=15<br>http://factordb.com/index.php?showid=1100000003588407386&base=15<br>http://factordb.com/index.php?showid=1100000002632398579&base=15<br>http://factordb.com/index.php?showid=1100000002321033312&base=15<br>http://factordb.com/index.php?showid=1100000002391199877&base=15<br>http://factordb.com/index.php?showid=1100000003588407806&base=15<br>http://factordb.com/index.php?showid=1100000000851967288&base=15<nowiki/>||0||–|| |- ||16||2347||3₁₁₆₁₃₇AF<br>4₇₂₇₈₅DD<br>DB₃₂₂₃₄<br>D0B₁₇₈₀₄<br>5BC₃₇₀₀D<br>90₃₅₄₂91<br>300F₁₉₆₀AF<br>20₁₇₁₃321<br>F8₁₅₁₇F<br>FAF₁₀₆₂45||116139<br>72787<br>32235<br>17806<br>3703<br>3545<br>1965<br>1717<br>1519<br>1066||139845<br>87644<br>38815<br>21441<br>4459<br>4269<br>2366<br>2067<br>1830<br>1284||(16¹¹⁶¹³⁹+619)/5<br>(4×16⁷²⁷⁸⁷+2291)/15<br>(206×16³²²³⁴−11)/15<br>(3131×16¹⁷⁸⁰⁴−11)/15<br>(459×16³⁷⁰¹+1)/5<br>9×16³⁵⁴⁴+145<br>769×16¹⁹⁶²−81<br>2×16¹⁷¹⁶+801<br>(233×16¹⁵¹⁸+97)/15<br>251×16¹⁰⁶⁴−187||http://factordb.com/index.php?id=1100000003851731988&open=prime<br>http://factordb.com/index.php?id=1100000003615909841&open=prime<br>http://factordb.com/index.php?id=1100000002383583629&open=prime<br>http://factordb.com/index.php?id=1100000003589278511&open=prime<br>http://factordb.com/index.php?id=1100000000993764322&open=prime<br>http://factordb.com/index.php?id=1100000000633424191&open=prime<br>http://factordb.com/index.php?id=1100000003588368750&open=prime<br>http://factordb.com/index.php?id=1100000003588386735&open=prime<br>http://factordb.com/index.php?id=1100000000633744824&open=prime<br>http://factordb.com/index.php?id=1100000003588387610&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003851731988&base=16<br>http://factordb.com/index.php?showid=1100000003615909841&base=16<br>http://factordb.com/index.php?showid=1100000002383583629&base=16<br>http://factordb.com/index.php?showid=1100000003589278511&base=16<br>http://factordb.com/index.php?showid=1100000000993764322&base=16<br>http://factordb.com/index.php?showid=1100000000633424191&base=16<br>http://factordb.com/index.php?showid=1100000003588368750&base=16<br>http://factordb.com/index.php?showid=1100000003588386735&base=16<br>http://factordb.com/index.php?showid=1100000000633744824&base=16<br>http://factordb.com/index.php?showid=1100000003588387610&base=16<nowiki/>||0||–|| |- ||17||10415~10427||95F₁₉₈₈₅₅<br>B0₁₈₉₀₈₃DB<br>F70₁₈₆₇₆₇1<br>970₁₆₆₀₄₇1<br>510₁₂₄₀₇₄D<br>49₁₁₁₃₃₃<br>B₆₇₁₀₃2E<br>570₅₁₃₁₀1<br>E9B₄₄₇₃₂<br>D0GD₃₇₀₉₆||198857<br>189086<br>186770<br>166050<br>124077<br>111334<br>67105<br>51313<br>44734<br>37099||244684<br>232661<br>229811<br>204316<br>152670<br>136991<br>82570<br>63138<br>55043<br>45649||(2543×17¹⁹⁸⁸⁵⁵−15)/16<br>11×17¹⁸⁹⁰⁸⁵+232<br>262×17¹⁸⁶⁷⁶⁸+1<br>160×17¹⁶⁶⁰⁴⁸+1<br>86×17¹²⁴⁰⁷⁵+13<br>(73×17¹¹¹³³³−9)/16<br>(11×17⁶⁷¹⁰⁵−2411)/16<br>92×17⁵¹³¹¹+1<br>(3963×17⁴⁴⁷³²−11)/16<br>(60381×17³⁷⁰⁹⁶−13)/16||http://factordb.com/index.php?id=1100000008610514108&open=prime<br>http://factordb.com/index.php?id=1100000008610515753&open=prime<br>http://factordb.com/index.php?id=1100000000765961429&open=prime<br>http://factordb.com/index.php?id=1100000000765961411&open=prime<br>http://factordb.com/index.php?id=1100000008610516879&open=prime<br>http://factordb.com/index.php?id=1100000000808118219&open=prime<br>http://factordb.com/index.php?id=1100000003993647842&open=prime<br>http://factordb.com/index.php?id=1100000000765961389&open=prime<br>http://factordb.com/index.php?id=1100000003883765450&open=prime<br>http://factordb.com/index.php?id=1100000003848346668&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008610514108&base=17<br>http://factordb.com/index.php?showid=1100000008610515753&base=17<br>http://factordb.com/index.php?showid=1100000000765961429&base=17<br>http://factordb.com/index.php?showid=1100000000765961411&base=17<br>http://factordb.com/index.php?showid=1100000008610516879&base=17<br>http://factordb.com/index.php?showid=1100000000808118219&base=17<br>http://factordb.com/index.php?showid=1100000003993647842&base=17<br>http://factordb.com/index.php?showid=1100000000765961389&base=17<br>http://factordb.com/index.php?showid=1100000003883765450&base=17<br>http://factordb.com/index.php?showid=1100000003848346668&base=17<nowiki/>||12||200000|| |- ||18||549||C0₆₂₆₈C5<br>H₇₆₆FH<br>80₂₉₈B<br>C0₁₁₆F5<br>HD₉₃<br>GG0₃₀1<br>CF₃₀5<br>B₁₉6B<br>CCF₁₄5<br>7₁₄G7||6271<br>768<br>300<br>119<br>94<br>33<br>32<br>21<br>17<br>16||7872<br>965<br>377<br>150<br>118<br>42<br>41<br>27<br>22<br>20||12×18⁶²⁷⁰+221<br>18⁷⁶⁸−37<br>8×18²⁹⁹+11<br>12×18¹¹⁸+275<br>(302×18⁹³−13)/17<br>304×18³¹+1<br>(219×18³¹−185)/17<br>(11×18²¹−1541)/17<br>(3891×18¹⁵−185)/17<br>(7×18¹⁶+2747)/17||http://factordb.com/index.php?id=1100000003590442437&open=prime<br>http://factordb.com/index.php?id=1100000003590430490&open=prime<br>http://factordb.com/index.php?id=1100000002355574745&open=prime<br>http://factordb.com/index.php?id=1100000002632837015&open=ecm<br>http://factordb.com/index.php?id=1100000002321052894&open=ecm<br>http://factordb.com/index.php?id=1100000000819230161&open=ecm<br>http://factordb.com/index.php?id=1100000002631240657&open=ecm<br>http://factordb.com/index.php?id=1100000003590430474&open=ecm<br>http://factordb.com/index.php?id=1100000003590430470&open=ecm<br>http://factordb.com/index.php?id=1100000003590430465&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003590442437&base=18<br>http://factordb.com/index.php?showid=1100000003590430490&base=18<br>http://factordb.com/index.php?showid=1100000002355574745&base=18<br>http://factordb.com/index.php?showid=1100000002632837015&base=18<br>http://factordb.com/index.php?showid=1100000002321052894&base=18<br>http://factordb.com/index.php?showid=1100000000819230161&base=18<br>http://factordb.com/index.php?showid=1100000002631240657&base=18<br>http://factordb.com/index.php?showid=1100000003590430474&base=18<br>http://factordb.com/index.php?showid=1100000003590430470&base=18<br>http://factordb.com/index.php?showid=1100000003590430465&base=18<nowiki/>||0||–|| |- ||19||31417~31434||1E70₁₂₂₈₉₆1<br>40₁₂₁₈₄₆HB5<br>35₁₂₀₅₆₂<br>FH0H₁₁₂₆₅₉<br>FG6₁₁₀₉₈₄<br>H₈₆₂₉₁6<br>D90₇₃₀₄₆9<br>4F0₄₉₈₄₇6<br>2₄₈₂₂₄7<br>2₄₅₈₈₆7A||122900<br>121850<br>120563<br>112662<br>110986<br>86292<br>73049<br>49850<br>48225<br>45888||157158<br>155816<br>154170<br>144067<br>110347<br>141924<br>93412<br>63746<br>61667<br>58679||634×19¹²²⁸⁹⁷+1<br>4×19¹²¹⁸⁴⁹+6351<br>(59×19¹²⁰⁵⁶²−5)/18<br>(103301×19¹¹²⁶⁵⁹−17)/18<br>(904×19¹¹⁰⁹⁸⁴−1)/3<br>(17×19⁸⁶²⁹²−215)/18<br>256×19⁷³⁰⁴⁷+9<br>91×19⁴⁹⁸⁴⁸+6<br>(19⁴⁸²²⁵+44)/9<br>(19⁴⁵⁸⁸⁸+926)/9||http://factordb.com/index.php?id=1100000001582289581&open=prime<br>http://factordb.com/index.php?id=1100000008755307222&open=prime<br>http://factordb.com/index.php?id=1100000005513825027&open=prime<br>http://factordb.com/index.php?id=1100000008755311453&open=prime<br>http://factordb.com/index.php?id=1100000000808118212&open=prime<br>http://factordb.com/index.php?id=1100000004163040839&open=prime<br>http://factordb.com/index.php?id=1100000003998413751&open=prime<br>http://factordb.com/index.php?id=1100000000808118332&open=prime<br>http://factordb.com/index.php?id=1100000003949188041&open=prime<br>http://factordb.com/index.php?id=1100000003949189035&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000001582289581&base=19<br>http://factordb.com/index.php?showid=1100000008755307222&base=19<br>http://factordb.com/index.php?showid=1100000005513825027&base=19<br>http://factordb.com/index.php?showid=1100000008755311453&base=19<br>http://factordb.com/index.php?showid=1100000000808118212&base=19<br>http://factordb.com/index.php?showid=1100000004163040839&base=19<br>http://factordb.com/index.php?showid=1100000003998413751&base=19<br>http://factordb.com/index.php?showid=1100000000808118332&base=19<br>http://factordb.com/index.php?showid=1100000003949188041&base=19<br>http://factordb.com/index.php?showid=1100000003949189035&base=19<nowiki/>||17||200000|| |- ||20||3314||G0₆₂₆₉D<br>CD₂₄₄₉<br>50₁₁₆₃AJ<br>J₆₅₅05J<br>JCJ₆₂₉<br>E₅₆₆C7<br>3A₅₂₇3<br>G₄₄₇99<br>EC0₄₂₉7<br>40₃₈₇404B||6271<br>2450<br>1166<br>658<br>631<br>568<br>529<br>449<br>432<br>392||8159<br>3188<br>1517<br>857<br>821<br>739<br>688<br>585<br>562<br>510||16×20⁶²⁷⁰+13<br>(241×20²⁴⁴⁹−13)/19<br>5×20¹¹⁶⁵+219<br>20⁶⁵⁸−7881<br>393×20⁶²⁹−1<br>(14×20⁵⁶⁸−907)/19<br>(67×20⁵²⁸−143)/19<br>(16×20⁴⁴⁹−2809)/19<br>292×20⁴³⁰+7<br>4×20³⁹¹+32091||http://factordb.com/index.php?id=1100000003590539457&open=prime<br>http://factordb.com/index.php?id=1100000002325393915&open=prime<br>http://factordb.com/index.php?id=1100000003590502412&open=prime<br>http://factordb.com/index.php?id=1100000003590502490&open=prime<br>http://factordb.com/index.php?id=1100000001559454258&open=prime<br>http://factordb.com/index.php?id=1100000003590502516&open=prime<br>http://factordb.com/index.php?id=1100000003590502531&open=prime<br>http://factordb.com/index.php?id=1100000000840126753&open=prime<br>http://factordb.com/index.php?id=1100000002633348702&open=prime<br>http://factordb.com/index.php?id=1100000003590502563&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003590539457&base=20<br>http://factordb.com/index.php?showid=1100000002325393915&base=20<br>http://factordb.com/index.php?showid=1100000003590502412&base=20<br>http://factordb.com/index.php?showid=1100000003590502490&base=20<br>http://factordb.com/index.php?showid=1100000001559454258&base=20<br>http://factordb.com/index.php?showid=1100000003590502516&base=20<br>http://factordb.com/index.php?showid=1100000003590502531&base=20<br>http://factordb.com/index.php?showid=1100000000840126753&base=20<br>http://factordb.com/index.php?showid=1100000002633348702&base=20<br>http://factordb.com/index.php?showid=1100000003590502563&base=20<nowiki/>||0||–|| |- ||21||13386~13394||27₁₈₄₄₉₉9D<br>F9₁₇₈₇₇₁D<br>2FC₁₁₂₀₂₂A<br>7₁₀₈₄₅₀ID<br>40₄₇₃₃₃9G<br>B90₄₅₀₁₉E5<br>HD₃₇₄₁₄<br>BD₃₅₀₂₇B<br>990₃₃₂₃₉99H<br>5₃₀₆₀₆FEK||184502<br>178773<br>112025<br>108452<br>47336<br>45023<br>37415<br>35029<br>33244<br>30609||243952<br>236377<br>148121<br>143397<br>62588<br>59531<br>49471<br>46316<br>43956<br>40472||(47×21¹⁸⁴⁵⁰¹+953)/20<br>(309×21¹⁷⁸⁷⁷²+71)/20<br>(288×21¹¹²⁰²³−13)/5<br>(7×21¹⁰⁸⁴⁵²+4733)/20<br>4×21⁴⁷³³⁵+205<br>240×21⁴⁵⁰²¹+299<br>(353×21³⁷⁴¹⁴−13)/20<br>(233×21³⁵⁰²⁸−53)/20<br>198×21³³²⁴²+4175<br>(21³⁰⁶⁰⁹+18455)/4||http://factordb.com/index.php?id=1100000008700600990&open=prime<br>http://factordb.com/index.php?id=1100000008700596669&open=prime<br>http://factordb.com/index.php?id=1100000008700593358&open=prime<br>http://factordb.com/index.php?id=1100000008700586183&open=prime<br>http://factordb.com/index.php?id=1100000000808118331&open=prime<br>http://factordb.com/index.php?id=1100000003996110311&open=prime<br>http://factordb.com/index.php?id=1100000003996110479&open=prime<br>http://factordb.com/index.php?id=1100000003996110718&open=prime<br>http://factordb.com/index.php?id=1100000003996110944&open=prime<br>http://factordb.com/index.php?id=1100000003996111130&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008700600990&base=21<br>http://factordb.com/index.php?showid=1100000008700596669&base=21<br>http://factordb.com/index.php?showid=1100000008700593358&base=21<br>http://factordb.com/index.php?showid=1100000008700586183&base=21<br>http://factordb.com/index.php?showid=1100000000808118331&base=21<br>http://factordb.com/index.php?showid=1100000003996110311&base=21<br>http://factordb.com/index.php?showid=1100000003996110479&base=21<br>http://factordb.com/index.php?showid=1100000003996110718&base=21<br>http://factordb.com/index.php?showid=1100000003996110944&base=21<br>http://factordb.com/index.php?showid=1100000003996111130&base=21<nowiki/>||8||200000|| |- ||22||8003||BK₂₂₀₀₁5<br>7₃₈₁₅2L<br>L₂₃₈₅KE7<br>7₉₅₉K7<br>J0₇₆₇IGGJ<br>K0₇₆₀EC1<br>I₆₂₆AF<br>E60₄₉₆L<br>L₄₈₃G3<br>L0₄₅₄B63||22003<br>3817<br>2388<br>961<br>772<br>764<br>628<br>499<br>485<br>458||29538<br>5124<br>3206<br>1290<br>1037<br>1026<br>843<br>670<br>652<br>615||(251×22²²⁰⁰²−335)/21<br>(22³⁸¹⁷−289)/3<br>22²³⁸⁸−653<br>(22⁹⁶¹+857)/3<br>19×22⁷⁷¹+199779<br>20×22⁷⁶³+7041<br>(6×22⁶²⁸−1259)/7<br>314×22⁴⁹⁷+21<br>22⁴⁸⁵−129<br>21×22⁴⁵⁷+5459||http://factordb.com/index.php?id=1100000003594696838&open=prime<br>http://factordb.com/index.php?id=1100000003591359839&open=prime<br>http://factordb.com/index.php?id=1100000003591360774&open=prime<br>http://factordb.com/index.php?id=1100000003591361817&open=prime<br>http://factordb.com/index.php?id=1100000003591362567&open=prime<br>http://factordb.com/index.php?id=1100000000632724415&open=prime<br>http://factordb.com/index.php?id=1100000000632724334&open=prime<br>http://factordb.com/index.php?id=1100000000632703239&open=prime<br>http://factordb.com/index.php?id=1100000003591364730&open=prime<br>http://factordb.com/index.php?id=1100000003591365331&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003594696838&base=22<br>http://factordb.com/index.php?showid=1100000003591359839&base=22<br>http://factordb.com/index.php?showid=1100000003591360774&base=22<br>http://factordb.com/index.php?showid=1100000003591361817&base=22<br>http://factordb.com/index.php?showid=1100000003591362567&base=22<br>http://factordb.com/index.php?showid=1100000000632724415&base=22<br>http://factordb.com/index.php?showid=1100000000632724334&base=22<br>http://factordb.com/index.php?showid=1100000000632703239&base=22<br>http://factordb.com/index.php?showid=1100000003591364730&base=22<br>http://factordb.com/index.php?showid=1100000003591365331&base=22<nowiki/>||0||–|| |- ||23||65178~65265||B0₉₃₀₄₆FB<br>L₈₆₄₄₄D<br>AJ₈₁₀₆₅4<br>20₇₃₅₆₀98<br>J₆₈₂₁₇G4<br>D70₆₆₇₇₀B<br>5F₆₂₃₄₀6<br>A7M7₆₁₅₃₂<br>B30₆₁₁₃₆5<br>EJ₅₂₁₆₉||93049<br>86445<br>81067<br>73563<br>68219<br>66773<br>62342<br>61535<br>61139<br>52170||126708<br>117715<br>110391<br>100172<br>92896<br>90927<br>84893<br>83794<br>83255<br>71042||11×23⁹³⁰⁴⁸+356<br>(21×23⁸⁶⁴⁴⁵−197)/22<br>(239×23⁸¹⁰⁶⁶−349)/22<br>2×23⁷³⁵⁶²+215<br>(19×23⁶⁸²¹⁹−1867)/22<br>306×23⁶⁶⁷⁷¹+11<br>(125×23⁶²³⁴¹−213)/22<br>(120413×23⁶¹⁵³²−7)/22<br>256×23⁶¹¹³⁷+5<br>(327×23⁵²¹⁶⁹−19)/22||http://factordb.com/index.php?id=1100000004691540361&open=prime<br>http://factordb.com/index.php?id=1100000004691546739&open=prime<br>http://factordb.com/index.php?id=1100000004691548070&open=prime<br>http://factordb.com/index.php?id=1100000004691548569&open=prime<br>http://factordb.com/index.php?id=1100000004691549462&open=prime<br>http://factordb.com/index.php?id=1100000004691549803&open=prime<br>http://factordb.com/index.php?id=1100000004691551005&open=prime<br>http://factordb.com/index.php?id=1100000004691556967&open=prime<br>http://factordb.com/index.php?id=1100000004691557254&open=prime<br>http://factordb.com/index.php?id=1100000004691557548&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004691540361&base=23<br>http://factordb.com/index.php?showid=1100000004691546739&base=23<br>http://factordb.com/index.php?showid=1100000004691548070&base=23<br>http://factordb.com/index.php?showid=1100000004691548569&base=23<br>http://factordb.com/index.php?showid=1100000004691549462&base=23<br>http://factordb.com/index.php?showid=1100000004691549803&base=23<br>http://factordb.com/index.php?showid=1100000004691551005&base=23<br>http://factordb.com/index.php?showid=1100000004691556967&base=23<br>http://factordb.com/index.php?showid=1100000004691557254&base=23<br>http://factordb.com/index.php?showid=1100000004691557548&base=23<nowiki/>||87||100000|| |- ||24||3409||N00N₈₁₂₉LN<br>88N₅₉₅₁<br>A0₂₉₅₁8ID<br>D₂₆₉₈LD<br>N₂₆₄₄LLN<br>BC0₃₃₁B<br>20₃₁₃7<br>C7₂₉₈<br>D0₂₅₉KKD<br>I0₂₄₁I5||8134<br>5953<br>2955<br>2700<br>2647<br>334<br>315<br>299<br>263<br>244||11227<br>8216<br>4079<br>3727<br>3654<br>461<br>434<br>413<br>363<br>337||13249×24⁸¹³¹−49<br>201×24⁵⁹⁵¹−1<br>10×24²⁹⁵⁴+5053<br>(13×24²⁷⁰⁰+4403)/23<br>24²⁶⁴⁷−1201<br>276×24³³²+11<br>2×24³¹⁴+7<br>(283×24²⁹⁸−7)/23<br>13×24²⁶²+12013<br>18×24²⁴³+437||http://factordb.com/index.php?id=1100000003593391606&open=prime<br>http://factordb.com/index.php?id=1100000003593275880&open=prime<br>http://factordb.com/index.php?id=1100000003593269654&open=prime<br>http://factordb.com/index.php?id=1100000003593269876&open=prime<br>http://factordb.com/index.php?id=1100000003593270089&open=prime<br>http://factordb.com/index.php?id=1100000002633359842&open=prime<br>http://factordb.com/index.php?id=1100000002355610241&open=prime<br>http://factordb.com/index.php?id=1100000002326181235&open=prime<br>http://factordb.com/index.php?id=1100000003593270725&open=prime<br>http://factordb.com/index.php?id=1100000002633360037&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003593391606&base=24<br>http://factordb.com/index.php?showid=1100000003593275880&base=24<br>http://factordb.com/index.php?showid=1100000003593269654&base=24<br>http://factordb.com/index.php?showid=1100000003593269876&base=24<br>http://factordb.com/index.php?showid=1100000003593270089&base=24<br>http://factordb.com/index.php?showid=1100000002633359842&base=24<br>http://factordb.com/index.php?showid=1100000002355610241&base=24<br>http://factordb.com/index.php?showid=1100000002326181235&base=24<br>http://factordb.com/index.php?showid=1100000003593270725&base=24<br>http://factordb.com/index.php?showid=1100000002633360037&base=24<nowiki/>||0||–|| |- ||25||133639~133724||E₉₈₃₉₆FOO<br>1J710₉₆₂₇₂1<br>NB0₈₅₅₉₈5NH<br>D70₈₁₅₈₁JJ7<br>F0₈₀₀₅₄HL<br>J010₇₅₉₄₃E7<br>K₆₇₇₇₁5I<br>LO₆₆₃₇₇KC<br>KJD0₆₃₃₉₉1<br>70₆₀₈₉₂D711||98399<br>96277<br>85603<br>81586<br>80057<br>75948<br>67773<br>66380<br>63403<br>60897||137556<br>134589<br>119668<br>114053<br>111915<br>106171<br>94743<br>92796<br>88634<br>85130||(7×25⁹⁸³⁹⁹+10613)/12<br>27676×25⁹⁶²⁷³+1<br>586×25⁸⁵⁶⁰¹+3717<br>332×25⁸¹⁵⁸⁴+12357<br>15×25⁸⁰⁰⁵⁶+446<br>11876×25⁷⁵⁹⁴⁵+357<br>(5×25⁶⁷⁷⁷³−2267)/6<br>22×25⁶⁶³⁷⁹−113<br>12988×25⁶³⁴⁰⁰+1<br>7×25⁶⁰⁸⁹⁶+207526||http://factordb.com/index.php?id=1100000000808118215&open=prime<br>http://factordb.com/index.php?id=1100000003983674902&open=prime<br>http://factordb.com/index.php?id=1100000004909706420&open=prime<br>http://factordb.com/index.php?id=1100000004909733266&open=prime<br>http://factordb.com/index.php?id=1100000004909750102&open=prime<br>http://factordb.com/index.php?id=1100000004909770736&open=prime<br>http://factordb.com/index.php?id=1100000004586986394&open=prime<br>http://factordb.com/index.php?id=1100000000808118270&open=prime<br>http://factordb.com/index.php?id=1100000004586986664&open=prime<br>http://factordb.com/index.php?id=1100000004586986798&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118215&base=25<br>http://factordb.com/index.php?showid=1100000003983674902&base=25<br>http://factordb.com/index.php?showid=1100000004909706420&base=25<br>http://factordb.com/index.php?showid=1100000004909733266&base=25<br>http://factordb.com/index.php?showid=1100000004909750102&base=25<br>http://factordb.com/index.php?showid=1100000004909770736&base=25<br>http://factordb.com/index.php?showid=1100000004586986394&base=25<br>http://factordb.com/index.php?showid=1100000000808118270&base=25<br>http://factordb.com/index.php?showid=1100000004586986664&base=25<br>http://factordb.com/index.php?showid=1100000004586986798&base=25<nowiki/>||85||100000|| |- ||26||25256~25259||85M₁₉₇₀₆₀B<br>M0₆₁₁₈₆2BB<br>J0₄₄₃₀₃KCB<br>6K₂₃₃₀₀5<br>LD0₂₀₉₇₅7<br>7₂₀₂₇₉OL<br>5₁₉₃₉₁6F<br>9GDK₁₅₉₂₀P<br>M₈₇₇₂P<br>K0₄₃₆₄I5||197063<br>61190<br>44307<br>23302<br>20978<br>20281<br>19393<br>15924<br>8773<br>4367||278839<br>86583<br>62694<br>32972<br>29684<br>28697<br>27440<br>22532<br>12414<br>6180||(5347×26¹⁹⁷⁰⁶¹−297)/25<br>22×26⁶¹¹⁸⁹+1649<br>19×26⁴⁴³⁰⁶+13843<br>(34×26²³³⁰¹−79)/5<br>559×26²⁰⁹⁷⁶+7<br>(7×26²⁰²⁸¹+11393)/25<br>(26¹⁹³⁹³+179)/5<br>(32569×26¹⁵⁹²¹+21)/5<br>(22×26⁸⁷⁷³+53)/25<br>20×26⁴³⁶⁶+473||http://factordb.com/index.php?id=1100000008573990023&open=prime<br>http://factordb.com/index.php?id=1100000003968169875&open=prime<br>http://factordb.com/index.php?id=1100000003968156595&open=prime<br>http://factordb.com/index.php?id=1100000003892628745&open=prime<br>http://factordb.com/index.php?id=1100000003892628658&open=prime<br>http://factordb.com/index.php?id=1100000003892628605&open=prime<br>http://factordb.com/index.php?id=1100000003850151202&open=prime<br>http://factordb.com/index.php?id=1100000003850155316&open=prime<br>http://factordb.com/index.php?id=1100000000758011195&open=prime<br>http://factordb.com/index.php?id=1100000002634136508&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008573990023&base=26<br>http://factordb.com/index.php?showid=1100000003968169875&base=26<br>http://factordb.com/index.php?showid=1100000003968156595&base=26<br>http://factordb.com/index.php?showid=1100000003892628745&base=26<br>http://factordb.com/index.php?showid=1100000003892628658&base=26<br>http://factordb.com/index.php?showid=1100000003892628605&base=26<br>http://factordb.com/index.php?showid=1100000003850151202&base=26<br>http://factordb.com/index.php?showid=1100000003850155316&base=26<br>http://factordb.com/index.php?showid=1100000000758011195&base=26<br>http://factordb.com/index.php?showid=1100000002634136508&base=26<nowiki/>||3||200000|| |- ||27||102852~102896||CA0F₈₈₈₈₃A<br>GNN0₇₈₇₉₅N<br>O44L₆₆₀₁₆7<br>NJ0₆₄₃₆₉H<br>ME₄₉₆₄₀9G<br>PH0₄₇₈₉₀1<br>QF₄₇₁₆₅AF5<br>J0₄₀₇₉₁PD<br>510₃₉₁₆₄I07<br>NGN0₃₆₃₂₉N||88887<br>78799<br>66020<br>64372<br>49643<br>47893<br>47169<br>40794<br>39169<br>36333||127230<br>112790<br>94499<br>92140<br>71058<br>68553<br>67516<br>58391<br>56065<br>52006||(234483×27⁸⁸⁸⁸⁴−145)/26<br>12308×27⁷⁸⁷⁹⁶+23<br>(457829×27⁶⁶⁰¹⁷−385)/26<br>640×27⁶⁴³⁷⁰+17<br>(293×27⁴⁹⁶⁴²−1736)/13<br>692×27⁴⁷⁸⁹¹+1<br>(691×27⁴⁷¹⁶⁸−95045)/26<br>19×27⁴⁰⁷⁹³+688<br>136×27³⁹¹⁶⁷+13129<br>17222×27³⁶³³⁰+23||http://factordb.com/index.php?id=1100000000808118233&open=prime<br>http://factordb.com/index.php?id=1100000004681348398&open=prime<br>http://factordb.com/index.php?id=1100000004374140861&open=prime<br>http://factordb.com/index.php?id=1100000004374138999&open=prime<br>http://factordb.com/index.php?id=1100000000819229859&open=prime<br>http://factordb.com/index.php?id=1100000004102754118&open=prime<br>http://factordb.com/index.php?id=1100000004102755880&open=prime<br>http://factordb.com/index.php?id=1100000004102758254&open=prime<br>http://factordb.com/index.php?id=1100000004102875088&open=prime<br>http://factordb.com/index.php?id=1100000004103372866&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118233&base=27<br>http://factordb.com/index.php?showid=1100000004681348398&base=27<br>http://factordb.com/index.php?showid=1100000004374140861&base=27<br>http://factordb.com/index.php?showid=1100000004374138999&base=27<br>http://factordb.com/index.php?showid=1100000000819229859&base=27<br>http://factordb.com/index.php?showid=1100000004102754118&base=27<br>http://factordb.com/index.php?showid=1100000004102755880&base=27<br>http://factordb.com/index.php?showid=1100000004102758254&base=27<br>http://factordb.com/index.php?showid=1100000004102875088&base=27<br>http://factordb.com/index.php?showid=1100000004103372866&base=27<nowiki/>||44||100000|| |- ||28||25528~25529||O4O₉₄₅₃₅9<br>5OA₃₁₂₃₈F<br>N6₂₄₀₅₁LR<br>D0₅₂₆₇77D<br>QO₄₂₃₉69<br>5₃₇₄₆8P<br>G0₁₈₉₉AN<br>A₁₄₂₃6F<br>5I₁₃₇₀F<br>5₁₃₃₂P8P||94538<br>31241<br>24054<br>5271<br>4242<br>3748<br>1902<br>1425<br>1372<br>1335||136812<br>45210<br>34810<br>7628<br>6139<br>5424<br>2753<br>2062<br>1985<br>1932||(6092×28⁹⁴⁵³⁶−143)/9<br>(4438×28³¹²³⁹+125)/27<br>(209×28²⁴⁰⁵³+3967)/9<br>13×28⁵²⁷⁰+5697<br>(242×28⁴²⁴¹−4679)/9<br>(5×28³⁷⁴⁸+2803)/27<br>16×28¹⁹⁰¹+303<br>(10×28¹⁴²⁵−2899)/27<br>(17×28¹³⁷¹−11)/3<br>(5×28¹³³⁵+426163)/27||http://factordb.com/index.php?id=1100000000808118231&open=prime<br>http://factordb.com/index.php?id=1100000003880455200&open=prime<br>http://factordb.com/index.php?id=1100000003879667576&open=prime<br>http://factordb.com/index.php?id=1100000003850151420&open=prime<br>http://factordb.com/index.php?id=1100000000840839934&open=prime<br>http://factordb.com/index.php?id=1100000003850161974&open=prime<br>http://factordb.com/index.php?id=1100000003850161973&open=prime<br>http://factordb.com/index.php?id=1100000000840839947&open=prime<br>http://factordb.com/index.php?id=1100000003850161972&open=prime<br>http://factordb.com/index.php?id=1100000003850161965&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118231&base=28<br>http://factordb.com/index.php?showid=1100000003880455200&base=28<br>http://factordb.com/index.php?showid=1100000003879667576&base=28<br>http://factordb.com/index.php?showid=1100000003850151420&base=28<br>http://factordb.com/index.php?showid=1100000000840839934&base=28<br>http://factordb.com/index.php?showid=1100000003850161974&base=28<br>http://factordb.com/index.php?showid=1100000003850161973&base=28<br>http://factordb.com/index.php?showid=1100000000840839947&base=28<br>http://factordb.com/index.php?showid=1100000003850161972&base=28<br>http://factordb.com/index.php?showid=1100000003850161965&base=28<nowiki/>||1||900000|| |- ||29||355242~355367||830₉₉₃₇₇4<br>GP5J₉₄₉₃₅<br>P05J₉₀₂₈₉<br>BBD0₈₈₈₈₈PB<br>8B₈₅₃₃₃G<br>L0₈₁₅₇₁5955<br>E0₇₇₃₇₂L7B<br>LPC₇₅₁₅₁9<br>JR0₇₄₆₂₂7<br>B₇₄₅₀₁0RP||99380<br>94938<br>90292<br>88893<br>85335<br>81576<br>77376<br>75154<br>74625<br>74504||145333<br>138837<br>132043<br>129997<br>124794<br>119297<br>113155<br>109905<br>109132<br>108955||235×29⁹⁹³⁷⁸+4<br>(397227×29⁹⁴⁹³⁵−19)/28<br>(588859×29⁹⁰²⁸⁹−19)/28<br>9583×29⁸⁸⁸⁹⁰+736<br>(235×29⁸⁵³³⁴+129)/14<br>21×29⁸¹⁵⁷⁵+129664<br>14×29⁷⁷³⁷⁵+17875<br>(4441×29⁷⁵¹⁵²−24)/7<br>578×29⁷⁴⁶²³+7<br>(11×29⁷⁴⁵⁰⁴−245655)/28||http://factordb.com/index.php?id=1100000008253882372&open=prime<br>http://factordb.com/index.php?id=1100000008253893542&open=prime<br>http://factordb.com/index.php?id=1100000008253899083&open=prime<br>http://factordb.com/index.php?id=1100000008253909183&open=prime<br>http://factordb.com/index.php?id=1100000008253921388&open=prime<br>http://factordb.com/index.php?id=1100000008253925955&open=prime<br>http://factordb.com/index.php?id=1100000008253931446&open=prime<br>http://factordb.com/index.php?id=1100000000808118236&open=prime<br>http://factordb.com/index.php?id=1100000008253934219&open=prime<br>http://factordb.com/index.php?id=1100000008253936120&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008253882372&base=29<br>http://factordb.com/index.php?showid=1100000008253893542&base=29<br>http://factordb.com/index.php?showid=1100000008253899083&base=29<br>http://factordb.com/index.php?showid=1100000008253909183&base=29<br>http://factordb.com/index.php?showid=1100000008253921388&base=29<br>http://factordb.com/index.php?showid=1100000008253925955&base=29<br>http://factordb.com/index.php?showid=1100000008253931446&base=29<br>http://factordb.com/index.php?showid=1100000000808118236&base=29<br>http://factordb.com/index.php?showid=1100000008253934219&base=29<br>http://factordb.com/index.php?showid=1100000008253936120&base=29<nowiki/>||125||100000|| |- ||30||2619||OT₃₄₂₀₅<br>I0₂₄₆₀₈D<br>5₄₈₈₂J<br>C0₁₀₂₂1<br>M0₅₄₇SS7<br>M₂₄₁QB<br>AN₂₀₆<br>50₁₆₄B<br>J₁₅₃QJ<br>J₉₄QQJ||34206<br>24610<br>4883<br>1024<br>551<br>243<br>207<br>166<br>155<br>97||50527<br>36352<br>7213<br>1513<br>814<br>359<br>306<br>245<br>229<br>144||25×30³⁴²⁰⁵−1<br>18×30²⁴⁶⁰⁹+13<br>(5×30⁴⁸⁸³+401)/29<br>12×30¹⁰²³+1<br>22×30⁵⁵⁰+26047<br>(22×30²⁴³+3139)/29<br>(313×30²⁰⁶−23)/29<br>5×30¹⁶⁵+11<br>(19×30¹⁵⁵+6071)/29<br>(19×30⁹⁷+188771)/29||http://factordb.com/index.php?id=1100000000800812865&open=prime<br>http://factordb.com/index.php?id=1100000003593967511&open=prime<br>http://factordb.com/index.php?id=1100000002327649423&open=prime<br>http://factordb.com/index.php?id=1100000000785448736&open=prime<br>http://factordb.com/index.php?id=1100000003593407988&open=prime<br>http://factordb.com/index.php?id=1100000003593408295&open=prime<br>http://factordb.com/index.php?id=1100000002327651073&open=prime<br>http://factordb.com/index.php?id=1100000002356282476&open=ecm<br>http://factordb.com/index.php?id=1100000003593409109&open=ecm<br>http://factordb.com/index.php?id=1100000003593409165&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000800812865&base=30<br>http://factordb.com/index.php?showid=1100000003593967511&base=30<br>http://factordb.com/index.php?showid=1100000002327649423&base=30<br>http://factordb.com/index.php?showid=1100000000785448736&base=30<br>http://factordb.com/index.php?showid=1100000003593407988&base=30<br>http://factordb.com/index.php?showid=1100000003593408295&base=30<br>http://factordb.com/index.php?showid=1100000002327651073&base=30<br>http://factordb.com/index.php?showid=1100000002356282476&base=30<br>http://factordb.com/index.php?showid=1100000003593409109&base=30<br>http://factordb.com/index.php?showid=1100000003593409165&base=30<nowiki/>||0||–|| |- ||31||569323~569400||2IIF₉₁₈₀₅<br>B0₈₈₃₀₉APO9<br>J0T₇₇₅₁₆<br>J090₇₇₁₂₈NNN<br>D₆₉₈₆₁QO<br>9MH0₆₈₆₃₇D<br>J₆₇₁₆₂D<br>N0₆₆₉₇₁32P<br>DDDQ0₆₄₀₈₈TD<br>U₆₃₈₆₁CM3||91808<br>88314<br>77518<br>77134<br>69863<br>68641<br>67163<br>66975<br>64094<br>63864||136918<br>131708<br>115608<br>115035<br>104191<br>102369<br>100165<br>99884<br>95587<br>95245||(4997×31⁹¹⁸⁰⁵−1)/2<br>11×31⁸⁸³¹³+322688<br>(17699×31⁷⁷⁵¹⁶−29)/30<br>18268×31⁷⁷¹³¹+22839<br>(13×31⁶⁹⁸⁶³+12407)/30<br>9348×31⁶⁸⁶³⁸+13<br>(19×31⁶⁷¹⁶³−199)/30<br>23×31⁶⁶⁹⁷⁴+2970<br>400205×31⁶⁴⁰⁹⁰+912<br>31⁶³⁸⁶⁴−17574||http://factordb.com/index.php?id=1100000007050395732&open=prime<br>http://factordb.com/index.php?id=1100000007050397309&open=prime<br>http://factordb.com/index.php?id=1100000007050398940&open=prime<br>http://factordb.com/index.php?id=1100000007050400178&open=prime<br>http://factordb.com/index.php?id=1100000006965878559&open=prime<br>http://factordb.com/index.php?id=1100000006965875678&open=prime<br>http://factordb.com/index.php?id=1100000006965873668&open=prime<br>http://factordb.com/index.php?id=1100000006965870538&open=prime<br>http://factordb.com/index.php?id=1100000006965868103&open=prime<br>http://factordb.com/index.php?id=1100000006965865343&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000007050395732&base=31<br>http://factordb.com/index.php?showid=1100000007050397309&base=31<br>http://factordb.com/index.php?showid=1100000007050398940&base=31<br>http://factordb.com/index.php?showid=1100000007050400178&base=31<br>http://factordb.com/index.php?showid=1100000006965878559&base=31<br>http://factordb.com/index.php?showid=1100000006965875678&base=31<br>http://factordb.com/index.php?showid=1100000006965873668&base=31<br>http://factordb.com/index.php?showid=1100000006965870538&base=31<br>http://factordb.com/index.php?showid=1100000006965868103&base=31<br>http://factordb.com/index.php?showid=1100000006965865343&base=31<nowiki/>||77||100000|| |- ||32||168882~169002||V₉₉₅₈₃63<br>6₈₉₀₇₄AF<br>8₇₇₇₀₀H<br>Q₇₇₄₀₁EQQQ3<br>8₇₇₂₄₉3<br>JM₇₆₀₂₈L<br>E₇₂₉₁₉IL<br>B0₆₇₆₈₀CB<br>GK₆₆₀₇₆F<br>KN₆₅₀₂₂||99585<br>89076<br>77701<br>77406<br>77250<br>76030<br>72921<br>67683<br>66078<br>65023||149891<br>134073<br>116952<br>116508<br>116273<br>114437<br>109757<br>101873<br>99458<br>97870||32⁹⁹⁵⁸⁵−829<br>(6×32⁸⁹⁰⁷⁶+4241)/31<br>(8×32⁷⁷⁷⁰¹+271)/31<br>(26×32⁷⁷⁴⁰⁶−390071011)/31<br>(8×32⁷⁷²⁵⁰−163)/31<br>(611×32⁷⁶⁰²⁹−53)/31<br>(14×32⁷²⁹²¹+4171)/31<br>11×32⁶⁷⁶⁸²+395<br>(516×32⁶⁶⁰⁷⁷−175)/31<br>(643×32⁶⁵⁰²²−23)/31||http://factordb.com/index.php?id=1100000005514892191&open=prime<br>http://factordb.com/index.php?id=1100000005514897129&open=prime<br>http://factordb.com/index.php?id=1100000005514901700&open=prime<br>http://factordb.com/index.php?id=1100000005514915338&open=prime<br>http://factordb.com/index.php?id=1100000005514918574&open=prime<br>http://factordb.com/index.php?id=1100000005514922523&open=prime<br>http://factordb.com/index.php?id=1100000004591654373&open=prime<br>http://factordb.com/index.php?id=1100000004591654467&open=prime<br>http://factordb.com/index.php?id=1100000004591654632&open=prime<br>http://factordb.com/index.php?id=1100000004591654952&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005514892191&base=32<br>http://factordb.com/index.php?showid=1100000005514897129&base=32<br>http://factordb.com/index.php?showid=1100000005514901700&base=32<br>http://factordb.com/index.php?showid=1100000005514915338&base=32<br>http://factordb.com/index.php?showid=1100000005514918574&base=32<br>http://factordb.com/index.php?showid=1100000005514922523&base=32<br>http://factordb.com/index.php?showid=1100000004591654373&base=32<br>http://factordb.com/index.php?showid=1100000004591654467&base=32<br>http://factordb.com/index.php?showid=1100000004591654632&base=32<br>http://factordb.com/index.php?showid=1100000004591654952&base=32<nowiki/>||120||100000|| |- ||33||280012~280093||DP₉₅₀₉₃M5<br>HJ0₉₄₂₉₅J<br>90₉₃₅₉₇Q<br>9F0₉₃₁₅₇N<br>7₈₉₄₄₉333H<br>K3₈₀₇₅₁6K<br>D₈₀₁₀₇9UD<br>VFU₇₂₂₀₄FK<br>J₆₈₇₁₅2BJ<br>DF0₆₈₃₆₇J||95096<br>94298<br>93599<br>93160<br>89453<br>80754<br>80110<br>72208<br>68718<br>68370||144405<br>143193<br>142131<br>141465<br>135835<br>122626<br>121648<br>109649<br>104350<br>103821||(441×33⁹⁵⁰⁹⁵−3833)/32<br>580×33⁹⁴²⁹⁶+19<br>9×33⁹³⁵⁹⁸+26<br>312×33⁹³¹⁵⁸+23<br>(7×33⁸⁹⁴⁵³−4743239)/32<br>(643×33⁸⁰⁷⁵³+3709)/32<br>(13×33⁸⁰¹¹⁰−121453)/32<br>(16623×33⁷²²⁰⁶−8095)/16<br>(19×33⁶⁸⁷¹⁸−600883)/32<br>444×33⁶⁸³⁶⁸+19||http://factordb.com/index.php?id=1100000005652348775&open=prime<br>http://factordb.com/index.php?id=1100000005652362811&open=prime<br>http://factordb.com/index.php?id=1100000005652375073&open=prime<br>http://factordb.com/index.php?id=1100000005652389776&open=prime<br>http://factordb.com/index.php?id=1100000005652430746&open=prime<br>http://factordb.com/index.php?id=1100000005652446200&open=prime<br>http://factordb.com/index.php?id=1100000005652461592&open=prime<br>http://factordb.com/index.php?id=1100000004614764298&open=prime<br>http://factordb.com/index.php?id=1100000004614770536&open=prime<br>http://factordb.com/index.php?id=1100000004614784274&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005652348775&base=33<br>http://factordb.com/index.php?showid=1100000005652362811&base=33<br>http://factordb.com/index.php?showid=1100000005652375073&base=33<br>http://factordb.com/index.php?showid=1100000005652389776&base=33<br>http://factordb.com/index.php?showid=1100000005652430746&base=33<br>http://factordb.com/index.php?showid=1100000005652446200&base=33<br>http://factordb.com/index.php?showid=1100000005652461592&base=33<br>http://factordb.com/index.php?showid=1100000004614764298&base=33<br>http://factordb.com/index.php?showid=1100000004614770536&base=33<br>http://factordb.com/index.php?showid=1100000004614784274&base=33<nowiki/>||81||100000|| |- ||34||184785~184832||GFGC₉₉₉₉₆5<br>90₉₇₉₅₀FJ<br>NM0₈₅₂₁₈KX<br>F₈₃₁₈₉H2HP<br>P₇₉₄₄₁444P<br>6₇₇₀₂₇8X<br>XQIQ₇₂₂₄₁D<br>T₆₆₅₃₀IF<br>4₆₆₁₅₂B<br>2EEC₆₆₀₃₉7||100000<br>97953<br>85222<br>83193<br>79445<br>77029<br>72245<br>66532<br>66153<br>66043||153148<br>150013<br>130516<br>127408<br>121669<br>117968<br>110642<br>101893<br>101312<br>101143||(209246×34⁹⁹⁹⁹⁷−81)/11<br>9×34⁹⁷⁹⁵²+529<br>804×34⁸⁵²²⁰+713<br>(5×34⁸³¹⁹³+700233)/11<br>(25×34⁷⁹⁴⁴⁵−28062367)/33<br>(2×34⁷⁷⁰²⁹+1043)/11<br>(1288676×34⁷²²⁴²−455)/33<br>(29×34⁶⁶⁵³²−12833)/33<br>(4×34⁶⁶¹⁵³+227)/33<br>(30826×34⁶⁶⁰⁴⁰−59)/11||http://factordb.com/index.php?id=1100000004702891268&open=prime<br>http://factordb.com/index.php?id=1100000004702894713&open=prime<br>http://factordb.com/index.php?id=1100000004702900996&open=prime<br>http://factordb.com/index.php?id=1100000004702910376&open=prime<br>http://factordb.com/index.php?id=1100000004702913746&open=prime<br>http://factordb.com/index.php?id=1100000004702918600&open=prime<br>http://factordb.com/index.php?id=1100000004399656529&open=prime<br>http://factordb.com/index.php?id=1100000004399657696&open=prime<br>http://factordb.com/index.php?id=1100000004399658651&open=prime<br>http://factordb.com/index.php?id=1100000004399659716&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004702891268&base=34<br>http://factordb.com/index.php?showid=1100000004702894713&base=34<br>http://factordb.com/index.php?showid=1100000004702900996&base=34<br>http://factordb.com/index.php?showid=1100000004702910376&base=34<br>http://factordb.com/index.php?showid=1100000004702913746&base=34<br>http://factordb.com/index.php?showid=1100000004702918600&base=34<br>http://factordb.com/index.php?showid=1100000004399656529&base=34<br>http://factordb.com/index.php?showid=1100000004399657696&base=34<br>http://factordb.com/index.php?showid=1100000004399658651&base=34<br>http://factordb.com/index.php?showid=1100000004399659716&base=34<nowiki/>||47||100000|| |- ||35||720002~720062||N0N₉₉₉₇₁9<br>V0₈₃₆₆₉E73<br>N₈₁₅₆₃K7N<br>BJ0₈₁₂₇₉N<br>J0₈₀₀₆₂FUH<br>43V₇₉₇₅₄<br>9₇₆₆₀₀K3<br>LB₇₁₃₆₆PB<br>Q₆₄₁₅₀H<br>50₆₃₃₉₇5R||99974<br>83673<br>81566<br>81282<br>80066<br>79756<br>76602<br>71369<br>64151<br>63400||154367<br>129197<br>125944<br>125505<br>123628<br>123148<br>118279<br>110199<br>99054<br>97894||(27393×35⁹⁹⁹⁷²−499)/34<br>31×35⁸³⁶⁷²+17398<br>(23×35⁸¹⁵⁶⁶−144013)/34<br>404×35⁸¹²⁸⁰+23<br>19×35⁸⁰⁰⁶⁵+19442<br>(4893×35⁷⁹⁷⁵⁴−31)/34<br>(9×35⁷⁶⁶⁰²+12877)/34<br>(725×35⁷¹³⁶⁸+16649)/34<br>(13×35⁶⁴¹⁵¹−166)/17<br>5×35⁶³³⁹⁹+202||http://factordb.com/index.php?id=1100000008248342445&open=prime<br>http://factordb.com/index.php?id=1100000008248353306&open=prime<br>http://factordb.com/index.php?id=1100000008248375642&open=prime<br>http://factordb.com/index.php?id=1100000008248397018&open=prime<br>http://factordb.com/index.php?id=1100000008248412468&open=prime<br>http://factordb.com/index.php?id=1100000008248418540&open=prime<br>http://factordb.com/index.php?id=1100000008248423670&open=prime<br>http://factordb.com/index.php?id=1100000008192119974&open=prime<br>http://factordb.com/index.php?id=1100000008192126630&open=prime<br>http://factordb.com/index.php?id=1100000008192129294&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008248342445&base=35<br>http://factordb.com/index.php?showid=1100000008248353306&base=35<br>http://factordb.com/index.php?showid=1100000008248375642&base=35<br>http://factordb.com/index.php?showid=1100000008248397018&base=35<br>http://factordb.com/index.php?showid=1100000008248412468&base=35<br>http://factordb.com/index.php?showid=1100000008248418540&base=35<br>http://factordb.com/index.php?showid=1100000008248423670&base=35<br>http://factordb.com/index.php?showid=1100000008192119974&base=35<br>http://factordb.com/index.php?showid=1100000008192126630&base=35<br>http://factordb.com/index.php?showid=1100000008192129294&base=35<nowiki/>||60||100000|| |- ||36||35286~35290||P₈₁₉₉₃SZ<br>S0₇₅₀₀₇8H<br>7K₂₆₅₆₇Z<br>J₁₀₁₁₇LJ<br>VL0₇₂₅₈J<br>EO0₆₁₇₇V<br>FZ₅₇₇₇3P<br>T09₄₆₁₈1<br>RY₄₅₆₂H<br>OZ₃₉₃₂AZ||81995<br>75010<br>26569<br>10119<br>7261<br>6180<br>5780<br>4621<br>4564<br>3935||127609<br>116739<br>41349<br>15748<br>11301<br>9618<br>8996<br>7192<br>7103<br>6124||(5×36⁸¹⁹⁹⁵+821)/7<br>28×36⁷⁵⁰⁰⁹+305<br>(53×36²⁶⁵⁶⁸+101)/7<br>(19×36¹⁰¹¹⁹+2501)/35<br>1137×36⁷²⁵⁹+19<br>528×36⁶¹⁷⁸+31<br>16×36⁵⁷⁷⁹−1163<br>(36549×36⁴⁶¹⁹−289)/35<br>(979×36⁴⁵⁶³−629)/35<br>25×36³⁹³⁴−901||http://factordb.com/index.php?id=1100000002394962083&open=prime<br>http://factordb.com/index.php?id=1100000004020085177&open=prime<br>http://factordb.com/index.php?id=1100000003896952461&open=prime<br>http://factordb.com/index.php?id=1100000003807362491&open=prime<br>http://factordb.com/index.php?id=1100000003807362489&open=prime<br>http://factordb.com/index.php?id=1100000003807362488&open=prime<br>http://factordb.com/index.php?id=1100000003807362487&open=prime<br>http://factordb.com/index.php?id=1100000003807362486&open=prime<br>http://factordb.com/index.php?id=1100000003807362485&open=prime<br>http://factordb.com/index.php?id=1100000000840634476&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000002394962083&base=36<br>http://factordb.com/index.php?showid=1100000004020085177&base=36<br>http://factordb.com/index.php?showid=1100000003896952461&base=36<br>http://factordb.com/index.php?showid=1100000003807362491&base=36<br>http://factordb.com/index.php?showid=1100000003807362489&base=36<br>http://factordb.com/index.php?showid=1100000003807362488&base=36<br>http://factordb.com/index.php?showid=1100000003807362487&base=36<br>http://factordb.com/index.php?showid=1100000003807362486&base=36<br>http://factordb.com/index.php?showid=1100000003807362485&base=36<br>http://factordb.com/index.php?showid=1100000000840634476&base=36<nowiki/>||4||200000|| |} == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] 20tz5s9ftgtoqqe207bljkjjv69ss05 2819705 2819703 2026-07-28T19:36:59Z Athene241 3100061 /* Solve the problem */ 2819705 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484; or for prime bases ''b'': https://oeis.org/A122396), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 11 family 2{5}: Always divisible by 2 or 3 * The base 14 family B{0}1: Always divisible by 3 or 5 * The base 13 family 95{0}3: Always divisible by 5, 7, or 17 * The base 16 family {4}D: Always divisible by 3, 7, or 13 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family {1}: Can be written as (9^''n''−1)/8 and can be factored as (3^''n''−1) × (3^''n''+1) / 8 * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 16 family 1{5}: Can be written as (4×16^''n''−1)/3 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) / 3 * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 27 family 7{Q}: Can be written as 8×27^''n''−1 and can be factored as (2×3^''n''−1) × (4×9^''n''+2×3^''n''+1) * The base 27 family 9{G}: Can be written as (125×27^''n''−8)/13 and can be factored as (5×3^''n''−2) × (25×9^''n''+10×3^''n''+4) * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 14 family {D}5: Can be written as 14^''n''−9, it is divisible by 5 if ''n'' is odd and can be factored as (14^''n''/2−3) × (14^''n''/2+3) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 17 family 7{9}: Can be written as (121×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17^''n''/2−3) × (11×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even * The base 19 family 7{2}: Can be written as (64×19^''n''−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19^''n''/2−1) × (8×19^''n''/2+1) / 9 if ''n'' is even * The base 24 family 3{N}: Can be written as 4×24^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24^''n''/2−1) × (2×24^''n''/2+1) if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored). The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]]. To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''^''n''+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1 we only used it to sieve the sequence ''a''×''b''^''n''+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''^''n''+''c'', but 2 may not divide (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the "easy" bases (bases ''b'' with ≤ 150 primes > 10²⁹⁹ (base ''b'' = 26 has 83 known primes > 10²⁹⁹ and 3 unsolved families, base ''b'' = 36 has 75 known primes > 10²⁹⁹ and 4 unsolved families, base ''b'' = 17 has 99 known primes > 10²⁹⁹ and 18 unsolved families, base ''b'' = 21 has 80 known primes > 10²⁹⁹ and 12 unsolved families, base ''b'' = 19 has 201 known primes > 10²⁹⁹ and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation. == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == Condensed table for bases 2 ≤ ''b'' ≤ 36 == {|class="wikitable" ||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write "''d''_''n''" if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)|| |- ||2||1||11||2||1||3||http://factordb.com/index.php?id=3&open=ecm||http://factordb.com/index.php?showid=3&base=2||0||–|| |- ||3||3||111<br>21<br>12||3<br>2<br>2||2<br>1<br>1||13<br>7<br>5||http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=13&base=3<br>http://factordb.com/index.php?showid=7&base=3<br>http://factordb.com/index.php?showid=5&base=3<nowiki/>||0||–|| |- ||4||5||221<br>31<br>23<br>13<br>11||3<br>2<br>2<br>2<br>2||2<br>2<br>2<br>1<br>1||41<br>13<br>11<br>7<br>5||http://factordb.com/index.php?id=41&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=41&base=4<br>http://factordb.com/index.php?showid=13&base=4<br>http://factordb.com/index.php?showid=11&base=4<br>http://factordb.com/index.php?showid=7&base=4<br>http://factordb.com/index.php?showid=5&base=4<nowiki/>||0||–|| |- ||5||22||10₉₃13<br>300031<br>44441<br>33331<br>33001<br>30301<br>14444<br>10103<br>3101<br>414||96<br>6<br>5<br>5<br>5<br>5<br>5<br>5<br>4<br>3||67<br>4<br>4<br>4<br>4<br>4<br>4<br>3<br>3<br>3||5⁹⁵+8<br>9391<br>3121<br>2341<br>2251<br>1951<br>1249<br>653<br>401<br>109||http://factordb.com/index.php?id=1100000000034686071&open=ecm<br>http://factordb.com/index.php?id=9391&open=ecm<br>http://factordb.com/index.php?id=3121&open=ecm<br>http://factordb.com/index.php?id=2341&open=ecm<br>http://factordb.com/index.php?id=2251&open=ecm<br>http://factordb.com/index.php?id=1951&open=ecm<br>http://factordb.com/index.php?id=1249&open=ecm<br>http://factordb.com/index.php?id=653&open=ecm<br>http://factordb.com/index.php?id=401&open=ecm<br>http://factordb.com/index.php?id=109&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000034686071&base=5<br>http://factordb.com/index.php?showid=9391&base=5<br>http://factordb.com/index.php?showid=3121&base=5<br>http://factordb.com/index.php?showid=2341&base=5<br>http://factordb.com/index.php?showid=2251&base=5<br>http://factordb.com/index.php?showid=1951&base=5<br>http://factordb.com/index.php?showid=1249&base=5<br>http://factordb.com/index.php?showid=653&base=5<br>http://factordb.com/index.php?showid=401&base=5<br>http://factordb.com/index.php?showid=109&base=5<nowiki/>||0||–|| |- ||6||11||40041<br>4441<br>4401<br>51<br>45<br>35<br>31<br>25<br>21<br>15||5<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||4<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||5209<br>1033<br>1009<br>31<br>29<br>23<br>19<br>17<br>13<br>11||http://factordb.com/index.php?id=5209&open=ecm<br>http://factordb.com/index.php?id=1033&open=ecm<br>http://factordb.com/index.php?id=1009&open=ecm<br>http://factordb.com/index.php?id=31&open=ecm<br>http://factordb.com/index.php?id=29&open=ecm<br>http://factordb.com/index.php?id=23&open=ecm<br>http://factordb.com/index.php?id=19&open=ecm<br>http://factordb.com/index.php?id=17&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<nowiki/>||http://factordb.com/index.php?showid=5209&base=6<br>http://factordb.com/index.php?showid=1033&base=6<br>http://factordb.com/index.php?showid=1009&base=6<br>http://factordb.com/index.php?showid=31&base=6<br>http://factordb.com/index.php?showid=29&base=6<br>http://factordb.com/index.php?showid=23&base=6<br>http://factordb.com/index.php?showid=19&base=6<br>http://factordb.com/index.php?showid=17&base=6<br>http://factordb.com/index.php?showid=13&base=6<br>http://factordb.com/index.php?showid=11&base=6<nowiki/>||0||–|| |- ||7||71||3₁₆1<br>510₇1<br>3₆01<br>1100021<br>531101<br>351101<br>300053<br>150001<br>100121<br>40054||17<br>10<br>8<br>7<br>6<br>6<br>6<br>6<br>6<br>5||15<br>9<br>7<br>6<br>5<br>5<br>5<br>5<br>5<br>4||(7¹⁷−5)/2<br>36×7⁸+1<br>(7⁸−47)/2<br>134471<br>91631<br>62819<br>50459<br>28813<br>16871<br>9643||http://factordb.com/index.php?id=116315256993601&open=ecm<br>http://factordb.com/index.php?id=207532837&open=ecm<br>http://factordb.com/index.php?id=2882377&open=ecm<br>http://factordb.com/index.php?id=134471&open=ecm<br>http://factordb.com/index.php?id=91631&open=ecm<br>http://factordb.com/index.php?id=62819&open=ecm<br>http://factordb.com/index.php?id=50459&open=ecm<br>http://factordb.com/index.php?id=28813&open=ecm<br>http://factordb.com/index.php?id=16871&open=ecm<br>http://factordb.com/index.php?id=9643&open=ecm<nowiki/>||http://factordb.com/index.php?showid=116315256993601&base=7<br>http://factordb.com/index.php?showid=207532837&base=7<br>http://factordb.com/index.php?showid=2882377&base=7<br>http://factordb.com/index.php?showid=134471&base=7<br>http://factordb.com/index.php?showid=91631&base=7<br>http://factordb.com/index.php?showid=62819&base=7<br>http://factordb.com/index.php?showid=50459&base=7<br>http://factordb.com/index.php?showid=28813&base=7<br>http://factordb.com/index.php?showid=16871&base=7<br>http://factordb.com/index.php?showid=9643&base=7<nowiki/>||0||–|| |- ||8||75||4₂₂₀7<br>5₁₃25<br>7₁₂1<br>77774₆1<br>74₇1<br>4₈1<br>5₅025<br>5550525<br>5500525<br>4₅77||221<br>15<br>13<br>11<br>9<br>9<br>8<br>7<br>7<br>7||200<br>14<br>12<br>10<br>9<br>8<br>8<br>7<br>7<br>7||(4×8²²¹+17)/7<br>(5×8¹⁵−173)/7<br>8¹³−7<br>(28669×8⁷−25)/7<br>(53×8⁸−25)/7<br>(4×8⁹−25)/7<br>(5×8⁸−2413)/7<br>1495381<br>1474901<br>(4×8⁷+185)/7||http://factordb.com/index.php?id=1100000000416605822&open=ecm<br>http://factordb.com/index.php?id=25131694349141&open=ecm<br>http://factordb.com/index.php?id=549755813881&open=ecm<br>http://factordb.com/index.php?id=8589035809&open=ecm<br>http://factordb.com/index.php?id=127027489&open=ecm<br>http://factordb.com/index.php?id=76695841&open=ecm<br>http://factordb.com/index.php?id=11983381&open=ecm<br>http://factordb.com/index.php?id=1495381&open=ecm<br>http://factordb.com/index.php?id=1474901&open=ecm<br>http://factordb.com/index.php?id=1198399&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000416605822&base=8<br>http://factordb.com/index.php?showid=25131694349141&base=8<br>http://factordb.com/index.php?showid=549755813881&base=8<br>http://factordb.com/index.php?showid=8589035809&base=8<br>http://factordb.com/index.php?showid=127027489&base=8<br>http://factordb.com/index.php?showid=76695841&base=8<br>http://factordb.com/index.php?showid=11983381&base=8<br>http://factordb.com/index.php?showid=1495381&base=8<br>http://factordb.com/index.php?showid=1474901&base=8<br>http://factordb.com/index.php?showid=1198399&base=8<nowiki/>||0||–|| |- ||9||151||30₁₁₅₈11<br>27₆₈₆07<br>76₃₂₉2<br>561₃₆<br>10₂₅57<br>30₂₀51<br>8₁₉335<br>727₁₅07<br>51₁₃61<br>10₁₁507||1161<br>689<br>331<br>38<br>28<br>23<br>22<br>19<br>16<br>15||1108<br>657<br>316<br>37<br>26<br>22<br>21<br>19<br>16<br>14||3×9¹¹⁶⁰+10<br>(23×9⁶⁸⁸−511)/8<br>(31×9³³⁰−19)/4<br>(409×9³⁶−1)/8<br>9²⁷+52<br>3×9²²+46<br>9²²−454<br>(527×9¹⁷−511)/8<br>(41×9¹⁵+359)/8<br>9¹⁴+412||http://factordb.com/index.php?id=1100000002376318423&open=prime<br>http://factordb.com/index.php?id=1100000002495467486&open=prime<br>http://factordb.com/index.php?id=1100000002359003642&open=prime<br>http://factordb.com/index.php?id=1100000001554010824&open=ecm<br>http://factordb.com/index.php?id=1100000002512830927&open=ecm<br>http://factordb.com/index.php?id=1100000000032261811&open=ecm<br>http://factordb.com/index.php?id=1100000002495736583&open=ecm<br>http://factordb.com/index.php?id=1100000003446800389&open=ecm<br>http://factordb.com/index.php?id=1055192051985121&open=ecm<br>http://factordb.com/index.php?id=22876792455373&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002376318423&base=9<br>http://factordb.com/index.php?showid=1100000002495467486&base=9<br>http://factordb.com/index.php?showid=1100000002359003642&base=9<br>http://factordb.com/index.php?showid=1100000001554010824&base=9<br>http://factordb.com/index.php?showid=1100000002512830927&base=9<br>http://factordb.com/index.php?showid=1100000000032261811&base=9<br>http://factordb.com/index.php?showid=1100000002495736583&base=9<br>http://factordb.com/index.php?showid=1100000003446800389&base=9<br>http://factordb.com/index.php?showid=1055192051985121&base=9<br>http://factordb.com/index.php?showid=22876792455373&base=9<nowiki/>||0||–|| |- ||10||77||50₂₈27<br>5₁₁1<br>805₅1<br>66600049<br>66000049<br>60₅49<br>220₅1<br>5200007<br>946669<br>666649||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||5×10³⁰+27<br>(5×10¹²−41)/9<br>(725×10⁶−41)/9<br>66600049<br>66000049<br>6×10⁷+49<br>22×10⁶+1<br>5200007<br>946669<br>666649||http://factordb.com/index.php?id=1100000000204142046&open=ecm<br>http://factordb.com/index.php?id=555555555551&open=ecm<br>http://factordb.com/index.php?id=80555551&open=ecm<br>http://factordb.com/index.php?id=66600049&open=ecm<br>http://factordb.com/index.php?id=66000049&open=ecm<br>http://factordb.com/index.php?id=60000049&open=ecm<br>http://factordb.com/index.php?id=22000001&open=ecm<br>http://factordb.com/index.php?id=5200007&open=ecm<br>http://factordb.com/index.php?id=946669&open=ecm<br>http://factordb.com/index.php?id=666649&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000204142046&base=10<br>http://factordb.com/index.php?showid=555555555551&base=10<br>http://factordb.com/index.php?showid=80555551&base=10<br>http://factordb.com/index.php?showid=66600049&base=10<br>http://factordb.com/index.php?showid=66000049&base=10<br>http://factordb.com/index.php?showid=60000049&base=10<br>http://factordb.com/index.php?showid=22000001&base=10<br>http://factordb.com/index.php?showid=5200007&base=10<br>http://factordb.com/index.php?showid=946669&base=10<br>http://factordb.com/index.php?showid=666649&base=10<nowiki/>||0||–|| |- ||11||1068||57₆₂₆₆₈<br>557₁₀₁₁<br>7₇₅₉44<br>A₇₁₃58<br>85₂₂₀05<br>507₂₀₆<br>5₁₆₁2A<br>50₁₂₆57<br>10₁₂₅51<br>326₁₂₂||62669<br>1013<br>761<br>715<br>223<br>208<br>163<br>129<br>128<br>124||65263<br>1055<br>793<br>745<br>233<br>217<br>170<br>134<br>133<br>129||(57×11⁶²⁶⁶⁸−7)/10<br>(607×11¹⁰¹¹−7)/10<br>(7×11⁷⁶¹−367)/10<br>11⁷¹⁵−58<br>(17×11²²²−111)/2<br>(557×11²⁰⁶−7)/10<br>(11¹⁶³−57)/2<br>5×11¹²⁸+62<br>11¹²⁷+56<br>(178×11¹²²−3)/5||http://factordb.com/index.php?id=1100000003573679860&open=prime<br>http://factordb.com/index.php?id=1100000002361376522&open=prime<br>http://factordb.com/index.php?id=1100000002505568840&open=prime<br>http://factordb.com/index.php?id=1100000003576826487&open=prime<br>http://factordb.com/index.php?id=1100000003576826769&open=ecm<br>http://factordb.com/index.php?id=1100000002518512744&open=ecm<br>http://factordb.com/index.php?id=1100000002391585327&open=ecm<br>http://factordb.com/index.php?id=1100000002632393378&open=ecm<br>http://factordb.com/index.php?id=1100000002391531300&open=ecm<br>http://factordb.com/index.php?id=1100000003576826781&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003573679860&base=11<br>http://factordb.com/index.php?showid=1100000002361376522&base=11<br>http://factordb.com/index.php?showid=1100000002505568840&base=11<br>http://factordb.com/index.php?showid=1100000003576826487&base=11<br>http://factordb.com/index.php?showid=1100000003576826769&base=11<br>http://factordb.com/index.php?showid=1100000002518512744&base=11<br>http://factordb.com/index.php?showid=1100000002391585327&base=11<br>http://factordb.com/index.php?showid=1100000002632393378&base=11<br>http://factordb.com/index.php?showid=1100000002391531300&base=11<br>http://factordb.com/index.php?showid=1100000003576826781&base=11<nowiki/>||0||–|| |- ||12||106||40₃₉77<br>B0₂₇9B<br>B₆99B<br>AA0₅1<br>B00099B<br>AAA0001<br>BBBAA1<br>A00065<br>44AAA1<br>BBBB1||42<br>30<br>9<br>8<br>7<br>7<br>6<br>6<br>6<br>5||45<br>33<br>10<br>9<br>8<br>8<br>7<br>7<br>7<br>6||4×12⁴¹+91<br>11×12²⁹+119<br>12⁹−313<br>130×12⁶+1<br>32847239<br>32555521<br>2985817<br>2488397<br>1097113<br>248821||http://factordb.com/index.php?id=1100000002375054575&open=ecm<br>http://factordb.com/index.php?id=1100000002354113100&open=ecm<br>http://factordb.com/index.php?id=5159780039&open=ecm<br>http://factordb.com/index.php?id=388177921&open=ecm<br>http://factordb.com/index.php?id=32847239&open=ecm<br>http://factordb.com/index.php?id=32555521&open=ecm<br>http://factordb.com/index.php?id=2985817&open=ecm<br>http://factordb.com/index.php?id=2488397&open=ecm<br>http://factordb.com/index.php?id=1097113&open=ecm<br>http://factordb.com/index.php?id=248821&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002375054575&base=12<br>http://factordb.com/index.php?showid=1100000002354113100&base=12<br>http://factordb.com/index.php?showid=5159780039&base=12<br>http://factordb.com/index.php?showid=388177921&base=12<br>http://factordb.com/index.php?showid=32847239&base=12<br>http://factordb.com/index.php?showid=32555521&base=12<br>http://factordb.com/index.php?showid=2985817&base=12<br>http://factordb.com/index.php?showid=2488397&base=12<br>http://factordb.com/index.php?showid=1097113&base=12<br>http://factordb.com/index.php?showid=248821&base=12<nowiki/>||0||–|| |- ||13||3197||A3₅₉₂₁₉₇A<br>95₁₉₇₄₂₀<br>80₃₂₀₁₇111<br>C5₂₃₇₅₅C<br>C₁₀₆₃₁92<br>B0₆₅₄₀BBA<br>390₆₂₆₆1<br>1770₂₇₀₃17<br>720₂₂₉₇2<br>930₁₅₅₁1||592199<br>197421<br>32021<br>23757<br>10633<br>6544<br>6269<br>2708<br>2300<br>1554||659677<br>219916<br>35670<br>26464<br>11845<br>7290<br>6983<br>3016<br>2562<br>1731||(41×13⁵⁹²¹⁹⁸+27)/4<br>(113×13¹⁹⁷⁴²⁰−5)/12<br>8×13³²⁰²⁰+183<br>(149×13²³⁷⁵⁶+79)/12<br>13¹⁰⁶³³−50<br>11×13⁶⁵⁴³+2012<br>48×13⁶²⁶⁷+1<br>267×13²⁷⁰⁵+20<br>93×13²²⁹⁸+2<br>120×13¹⁵⁵²+1||http://factordb.com/index.php?id=1100000005489162806&open=prime<br>http://factordb.com/index.php?id=1100000003943359311&open=prime<br>http://factordb.com/index.php?id=1100000000490878060&open=prime<br>http://factordb.com/index.php?id=1100000003590647776&open=prime<br>http://factordb.com/index.php?id=1100000003590493750&open=prime<br>http://factordb.com/index.php?id=1100000002616382906&open=prime<br>http://factordb.com/index.php?id=1100000000765961441&open=prime<br>http://factordb.com/index.php?id=1100000003590430825&open=prime<br>http://factordb.com/index.php?id=1100000002632396910&open=prime<br>http://factordb.com/index.php?id=1100000000765961452&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005489162806&base=13<br>http://factordb.com/index.php?showid=1100000003943359311&base=13<br>http://factordb.com/index.php?showid=1100000000490878060&base=13<br>http://factordb.com/index.php?showid=1100000003590647776&base=13<br>http://factordb.com/index.php?showid=1100000003590493750&base=13<br>http://factordb.com/index.php?showid=1100000002616382906&base=13<br>http://factordb.com/index.php?showid=1100000000765961441&base=13<br>http://factordb.com/index.php?showid=1100000003590430825&base=13<br>http://factordb.com/index.php?showid=1100000002632396910&base=13<br>http://factordb.com/index.php?showid=1100000000765961452&base=13<nowiki/>||0||–|| |- ||14||650||4D₁₉₆₉₈<br>34D₇₀₈<br>8D₁₄₁85<br>8₈₆B<br>40₈₃49<br>8C₇₉3<br>18₇₉B<br>6B₇₇2B<br>4₆₃09<br>A₅₉3||19699<br>710<br>144<br>87<br>86<br>81<br>81<br>80<br>65<br>60||22578<br>814<br>165<br>100<br>99<br>93<br>92<br>92<br>74<br>69||5×14¹⁹⁶⁹⁸−1<br>47×14⁷⁰⁸−1<br>9×14¹⁴³−79<br>(8×14⁸⁷+31)/13<br>4×14⁸⁵+65<br>(116×14⁸⁰−129)/13<br>(21×14⁸⁰+31)/13<br>(89×14⁷⁹−1649)/13<br>(4×14⁶⁵−667)/13<br>(10×14⁶⁰−101)/13||http://factordb.com/index.php?id=1100000000884560233&open=prime<br>http://factordb.com/index.php?id=1100000001540144903&open=prime<br>http://factordb.com/index.php?id=1100000003575856650&open=ecm<br>http://factordb.com/index.php?id=1100000002321014379&open=ecm<br>http://factordb.com/index.php?id=1100000000823937973&open=ecm<br>http://factordb.com/index.php?id=1100000002631073246&open=ecm<br>http://factordb.com/index.php?id=1100000002384401372&open=ecm<br>http://factordb.com/index.php?id=1100000002631077787&open=ecm<br>http://factordb.com/index.php?id=1100000000840126683&open=ecm<br>http://factordb.com/index.php?id=1100000002321038522&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000884560233&base=14<br>http://factordb.com/index.php?showid=1100000001540144903&base=14<br>http://factordb.com/index.php?showid=1100000003575856650&base=14<br>http://factordb.com/index.php?showid=1100000002321014379&base=14<br>http://factordb.com/index.php?showid=1100000000823937973&base=14<br>http://factordb.com/index.php?showid=1100000002631073246&base=14<br>http://factordb.com/index.php?showid=1100000002384401372&base=14<br>http://factordb.com/index.php?showid=1100000002631077787&base=14<br>http://factordb.com/index.php?showid=1100000000840126683&base=14<br>http://factordb.com/index.php?showid=1100000002321038522&base=14<nowiki/>||0||–|| |- ||15||1284||7₁₅₅97<br>E₁₄₅397<br>96₁₀₄08<br>7₇₃CE<br>7₅₉CCE<br>50₃₃17<br>EB₃₁<br>6330₂₆1<br>7050₂₄B<br>B70₂₄1||157<br>148<br>107<br>75<br>62<br>36<br>32<br>30<br>28<br>27||185<br>175<br>126<br>88<br>73<br>42<br>38<br>35<br>33<br>32||(15¹⁵⁷+59)/2<br>15¹⁴⁸−2558<br>(66×15¹⁰⁶−619)/7<br>(15⁷⁵+163)/2<br>(15⁶²+2413)/2<br>5×15³⁵+22<br>(207×15³¹−11)/14<br>1398×15²⁷+1<br>1580×15²⁵+11<br>172×15²⁵+1||http://factordb.com/index.php?id=1100000002454891840&open=ecm<br>http://factordb.com/index.php?id=1100000002454900849&open=ecm<br>http://factordb.com/index.php?id=1100000000823937997&open=ecm<br>http://factordb.com/index.php?id=1100000003588407143&open=ecm<br>http://factordb.com/index.php?id=1100000003588407386&open=ecm<br>http://factordb.com/index.php?id=1100000002632398579&open=ecm<br>http://factordb.com/index.php?id=1100000002321033312&open=ecm<br>http://factordb.com/index.php?id=1100000002391199877&open=ecm<br>http://factordb.com/index.php?id=1100000003588407806&open=ecm<br>http://factordb.com/index.php?id=1100000000851967288&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002454891840&base=15<br>http://factordb.com/index.php?showid=1100000002454900849&base=15<br>http://factordb.com/index.php?showid=1100000000823937997&base=15<br>http://factordb.com/index.php?showid=1100000003588407143&base=15<br>http://factordb.com/index.php?showid=1100000003588407386&base=15<br>http://factordb.com/index.php?showid=1100000002632398579&base=15<br>http://factordb.com/index.php?showid=1100000002321033312&base=15<br>http://factordb.com/index.php?showid=1100000002391199877&base=15<br>http://factordb.com/index.php?showid=1100000003588407806&base=15<br>http://factordb.com/index.php?showid=1100000000851967288&base=15<nowiki/>||0||–|| |- ||16||2347||3₁₁₆₁₃₇AF<br>4₇₂₇₈₅DD<br>DB₃₂₂₃₄<br>D0B₁₇₈₀₄<br>5BC₃₇₀₀D<br>90₃₅₄₂91<br>300F₁₉₆₀AF<br>20₁₇₁₃321<br>F8₁₅₁₇F<br>FAF₁₀₆₂45||116139<br>72787<br>32235<br>17806<br>3703<br>3545<br>1965<br>1717<br>1519<br>1066||139845<br>87644<br>38815<br>21441<br>4459<br>4269<br>2366<br>2067<br>1830<br>1284||(16¹¹⁶¹³⁹+619)/5<br>(4×16⁷²⁷⁸⁷+2291)/15<br>(206×16³²²³⁴−11)/15<br>(3131×16¹⁷⁸⁰⁴−11)/15<br>(459×16³⁷⁰¹+1)/5<br>9×16³⁵⁴⁴+145<br>769×16¹⁹⁶²−81<br>2×16¹⁷¹⁶+801<br>(233×16¹⁵¹⁸+97)/15<br>251×16¹⁰⁶⁴−187||http://factordb.com/index.php?id=1100000003851731988&open=prime<br>http://factordb.com/index.php?id=1100000003615909841&open=prime<br>http://factordb.com/index.php?id=1100000002383583629&open=prime<br>http://factordb.com/index.php?id=1100000003589278511&open=prime<br>http://factordb.com/index.php?id=1100000000993764322&open=prime<br>http://factordb.com/index.php?id=1100000000633424191&open=prime<br>http://factordb.com/index.php?id=1100000003588368750&open=prime<br>http://factordb.com/index.php?id=1100000003588386735&open=prime<br>http://factordb.com/index.php?id=1100000000633744824&open=prime<br>http://factordb.com/index.php?id=1100000003588387610&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003851731988&base=16<br>http://factordb.com/index.php?showid=1100000003615909841&base=16<br>http://factordb.com/index.php?showid=1100000002383583629&base=16<br>http://factordb.com/index.php?showid=1100000003589278511&base=16<br>http://factordb.com/index.php?showid=1100000000993764322&base=16<br>http://factordb.com/index.php?showid=1100000000633424191&base=16<br>http://factordb.com/index.php?showid=1100000003588368750&base=16<br>http://factordb.com/index.php?showid=1100000003588386735&base=16<br>http://factordb.com/index.php?showid=1100000000633744824&base=16<br>http://factordb.com/index.php?showid=1100000003588387610&base=16<nowiki/>||0||–|| |- ||17||10415~10427||95F₁₉₈₈₅₅<br>B0₁₈₉₀₈₃DB<br>F70₁₈₆₇₆₇1<br>970₁₆₆₀₄₇1<br>510₁₂₄₀₇₄D<br>49₁₁₁₃₃₃<br>B₆₇₁₀₃2E<br>570₅₁₃₁₀1<br>E9B₄₄₇₃₂<br>D0GD₃₇₀₉₆||198857<br>189086<br>186770<br>166050<br>124077<br>111334<br>67105<br>51313<br>44734<br>37099||244684<br>232661<br>229811<br>204316<br>152670<br>136991<br>82570<br>63138<br>55043<br>45649||(2543×17¹⁹⁸⁸⁵⁵−15)/16<br>11×17¹⁸⁹⁰⁸⁵+232<br>262×17¹⁸⁶⁷⁶⁸+1<br>160×17¹⁶⁶⁰⁴⁸+1<br>86×17¹²⁴⁰⁷⁵+13<br>(73×17¹¹¹³³³−9)/16<br>(11×17⁶⁷¹⁰⁵−2411)/16<br>92×17⁵¹³¹¹+1<br>(3963×17⁴⁴⁷³²−11)/16<br>(60381×17³⁷⁰⁹⁶−13)/16||http://factordb.com/index.php?id=1100000008610514108&open=prime<br>http://factordb.com/index.php?id=1100000008610515753&open=prime<br>http://factordb.com/index.php?id=1100000000765961429&open=prime<br>http://factordb.com/index.php?id=1100000000765961411&open=prime<br>http://factordb.com/index.php?id=1100000008610516879&open=prime<br>http://factordb.com/index.php?id=1100000000808118219&open=prime<br>http://factordb.com/index.php?id=1100000003993647842&open=prime<br>http://factordb.com/index.php?id=1100000000765961389&open=prime<br>http://factordb.com/index.php?id=1100000003883765450&open=prime<br>http://factordb.com/index.php?id=1100000003848346668&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008610514108&base=17<br>http://factordb.com/index.php?showid=1100000008610515753&base=17<br>http://factordb.com/index.php?showid=1100000000765961429&base=17<br>http://factordb.com/index.php?showid=1100000000765961411&base=17<br>http://factordb.com/index.php?showid=1100000008610516879&base=17<br>http://factordb.com/index.php?showid=1100000000808118219&base=17<br>http://factordb.com/index.php?showid=1100000003993647842&base=17<br>http://factordb.com/index.php?showid=1100000000765961389&base=17<br>http://factordb.com/index.php?showid=1100000003883765450&base=17<br>http://factordb.com/index.php?showid=1100000003848346668&base=17<nowiki/>||12||200000|| |- ||18||549||C0₆₂₆₈C5<br>H₇₆₆FH<br>80₂₉₈B<br>C0₁₁₆F5<br>HD₉₃<br>GG0₃₀1<br>CF₃₀5<br>B₁₉6B<br>CCF₁₄5<br>7₁₄G7||6271<br>768<br>300<br>119<br>94<br>33<br>32<br>21<br>17<br>16||7872<br>965<br>377<br>150<br>118<br>42<br>41<br>27<br>22<br>20||12×18⁶²⁷⁰+221<br>18⁷⁶⁸−37<br>8×18²⁹⁹+11<br>12×18¹¹⁸+275<br>(302×18⁹³−13)/17<br>304×18³¹+1<br>(219×18³¹−185)/17<br>(11×18²¹−1541)/17<br>(3891×18¹⁵−185)/17<br>(7×18¹⁶+2747)/17||http://factordb.com/index.php?id=1100000003590442437&open=prime<br>http://factordb.com/index.php?id=1100000003590430490&open=prime<br>http://factordb.com/index.php?id=1100000002355574745&open=prime<br>http://factordb.com/index.php?id=1100000002632837015&open=ecm<br>http://factordb.com/index.php?id=1100000002321052894&open=ecm<br>http://factordb.com/index.php?id=1100000000819230161&open=ecm<br>http://factordb.com/index.php?id=1100000002631240657&open=ecm<br>http://factordb.com/index.php?id=1100000003590430474&open=ecm<br>http://factordb.com/index.php?id=1100000003590430470&open=ecm<br>http://factordb.com/index.php?id=1100000003590430465&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003590442437&base=18<br>http://factordb.com/index.php?showid=1100000003590430490&base=18<br>http://factordb.com/index.php?showid=1100000002355574745&base=18<br>http://factordb.com/index.php?showid=1100000002632837015&base=18<br>http://factordb.com/index.php?showid=1100000002321052894&base=18<br>http://factordb.com/index.php?showid=1100000000819230161&base=18<br>http://factordb.com/index.php?showid=1100000002631240657&base=18<br>http://factordb.com/index.php?showid=1100000003590430474&base=18<br>http://factordb.com/index.php?showid=1100000003590430470&base=18<br>http://factordb.com/index.php?showid=1100000003590430465&base=18<nowiki/>||0||–|| |- ||19||31417~31434||1E70₁₂₂₈₉₆1<br>40₁₂₁₈₄₆HB5<br>35₁₂₀₅₆₂<br>FH0H₁₁₂₆₅₉<br>FG6₁₁₀₉₈₄<br>H₈₆₂₉₁6<br>D90₇₃₀₄₆9<br>4F0₄₉₈₄₇6<br>2₄₈₂₂₄7<br>2₄₅₈₈₆7A||122900<br>121850<br>120563<br>112662<br>110986<br>86292<br>73049<br>49850<br>48225<br>45888||157158<br>155816<br>154170<br>144067<br>110347<br>141924<br>93412<br>63746<br>61667<br>58679||634×19¹²²⁸⁹⁷+1<br>4×19¹²¹⁸⁴⁹+6351<br>(59×19¹²⁰⁵⁶²−5)/18<br>(103301×19¹¹²⁶⁵⁹−17)/18<br>(904×19¹¹⁰⁹⁸⁴−1)/3<br>(17×19⁸⁶²⁹²−215)/18<br>256×19⁷³⁰⁴⁷+9<br>91×19⁴⁹⁸⁴⁸+6<br>(19⁴⁸²²⁵+44)/9<br>(19⁴⁵⁸⁸⁸+926)/9||http://factordb.com/index.php?id=1100000001582289581&open=prime<br>http://factordb.com/index.php?id=1100000008755307222&open=prime<br>http://factordb.com/index.php?id=1100000005513825027&open=prime<br>http://factordb.com/index.php?id=1100000008755311453&open=prime<br>http://factordb.com/index.php?id=1100000000808118212&open=prime<br>http://factordb.com/index.php?id=1100000004163040839&open=prime<br>http://factordb.com/index.php?id=1100000003998413751&open=prime<br>http://factordb.com/index.php?id=1100000000808118332&open=prime<br>http://factordb.com/index.php?id=1100000003949188041&open=prime<br>http://factordb.com/index.php?id=1100000003949189035&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000001582289581&base=19<br>http://factordb.com/index.php?showid=1100000008755307222&base=19<br>http://factordb.com/index.php?showid=1100000005513825027&base=19<br>http://factordb.com/index.php?showid=1100000008755311453&base=19<br>http://factordb.com/index.php?showid=1100000000808118212&base=19<br>http://factordb.com/index.php?showid=1100000004163040839&base=19<br>http://factordb.com/index.php?showid=1100000003998413751&base=19<br>http://factordb.com/index.php?showid=1100000000808118332&base=19<br>http://factordb.com/index.php?showid=1100000003949188041&base=19<br>http://factordb.com/index.php?showid=1100000003949189035&base=19<nowiki/>||17||200000|| |- ||20||3314||G0₆₂₆₉D<br>CD₂₄₄₉<br>50₁₁₆₃AJ<br>J₆₅₅05J<br>JCJ₆₂₉<br>E₅₆₆C7<br>3A₅₂₇3<br>G₄₄₇99<br>EC0₄₂₉7<br>40₃₈₇404B||6271<br>2450<br>1166<br>658<br>631<br>568<br>529<br>449<br>432<br>392||8159<br>3188<br>1517<br>857<br>821<br>739<br>688<br>585<br>562<br>510||16×20⁶²⁷⁰+13<br>(241×20²⁴⁴⁹−13)/19<br>5×20¹¹⁶⁵+219<br>20⁶⁵⁸−7881<br>393×20⁶²⁹−1<br>(14×20⁵⁶⁸−907)/19<br>(67×20⁵²⁸−143)/19<br>(16×20⁴⁴⁹−2809)/19<br>292×20⁴³⁰+7<br>4×20³⁹¹+32091||http://factordb.com/index.php?id=1100000003590539457&open=prime<br>http://factordb.com/index.php?id=1100000002325393915&open=prime<br>http://factordb.com/index.php?id=1100000003590502412&open=prime<br>http://factordb.com/index.php?id=1100000003590502490&open=prime<br>http://factordb.com/index.php?id=1100000001559454258&open=prime<br>http://factordb.com/index.php?id=1100000003590502516&open=prime<br>http://factordb.com/index.php?id=1100000003590502531&open=prime<br>http://factordb.com/index.php?id=1100000000840126753&open=prime<br>http://factordb.com/index.php?id=1100000002633348702&open=prime<br>http://factordb.com/index.php?id=1100000003590502563&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003590539457&base=20<br>http://factordb.com/index.php?showid=1100000002325393915&base=20<br>http://factordb.com/index.php?showid=1100000003590502412&base=20<br>http://factordb.com/index.php?showid=1100000003590502490&base=20<br>http://factordb.com/index.php?showid=1100000001559454258&base=20<br>http://factordb.com/index.php?showid=1100000003590502516&base=20<br>http://factordb.com/index.php?showid=1100000003590502531&base=20<br>http://factordb.com/index.php?showid=1100000000840126753&base=20<br>http://factordb.com/index.php?showid=1100000002633348702&base=20<br>http://factordb.com/index.php?showid=1100000003590502563&base=20<nowiki/>||0||–|| |- ||21||13386~13394||27₁₈₄₄₉₉9D<br>F9₁₇₈₇₇₁D<br>2FC₁₁₂₀₂₂A<br>7₁₀₈₄₅₀ID<br>40₄₇₃₃₃9G<br>B90₄₅₀₁₉E5<br>HD₃₇₄₁₄<br>BD₃₅₀₂₇B<br>990₃₃₂₃₉99H<br>5₃₀₆₀₆FEK||184502<br>178773<br>112025<br>108452<br>47336<br>45023<br>37415<br>35029<br>33244<br>30609||243952<br>236377<br>148121<br>143397<br>62588<br>59531<br>49471<br>46316<br>43956<br>40472||(47×21¹⁸⁴⁵⁰¹+953)/20<br>(309×21¹⁷⁸⁷⁷²+71)/20<br>(288×21¹¹²⁰²³−13)/5<br>(7×21¹⁰⁸⁴⁵²+4733)/20<br>4×21⁴⁷³³⁵+205<br>240×21⁴⁵⁰²¹+299<br>(353×21³⁷⁴¹⁴−13)/20<br>(233×21³⁵⁰²⁸−53)/20<br>198×21³³²⁴²+4175<br>(21³⁰⁶⁰⁹+18455)/4||http://factordb.com/index.php?id=1100000008700600990&open=prime<br>http://factordb.com/index.php?id=1100000008700596669&open=prime<br>http://factordb.com/index.php?id=1100000008700593358&open=prime<br>http://factordb.com/index.php?id=1100000008700586183&open=prime<br>http://factordb.com/index.php?id=1100000000808118331&open=prime<br>http://factordb.com/index.php?id=1100000003996110311&open=prime<br>http://factordb.com/index.php?id=1100000003996110479&open=prime<br>http://factordb.com/index.php?id=1100000003996110718&open=prime<br>http://factordb.com/index.php?id=1100000003996110944&open=prime<br>http://factordb.com/index.php?id=1100000003996111130&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008700600990&base=21<br>http://factordb.com/index.php?showid=1100000008700596669&base=21<br>http://factordb.com/index.php?showid=1100000008700593358&base=21<br>http://factordb.com/index.php?showid=1100000008700586183&base=21<br>http://factordb.com/index.php?showid=1100000000808118331&base=21<br>http://factordb.com/index.php?showid=1100000003996110311&base=21<br>http://factordb.com/index.php?showid=1100000003996110479&base=21<br>http://factordb.com/index.php?showid=1100000003996110718&base=21<br>http://factordb.com/index.php?showid=1100000003996110944&base=21<br>http://factordb.com/index.php?showid=1100000003996111130&base=21<nowiki/>||8||200000|| |- ||22||8003||BK₂₂₀₀₁5<br>7₃₈₁₅2L<br>L₂₃₈₅KE7<br>7₉₅₉K7<br>J0₇₆₇IGGJ<br>K0₇₆₀EC1<br>I₆₂₆AF<br>E60₄₉₆L<br>L₄₈₃G3<br>L0₄₅₄B63||22003<br>3817<br>2388<br>961<br>772<br>764<br>628<br>499<br>485<br>458||29538<br>5124<br>3206<br>1290<br>1037<br>1026<br>843<br>670<br>652<br>615||(251×22²²⁰⁰²−335)/21<br>(22³⁸¹⁷−289)/3<br>22²³⁸⁸−653<br>(22⁹⁶¹+857)/3<br>19×22⁷⁷¹+199779<br>20×22⁷⁶³+7041<br>(6×22⁶²⁸−1259)/7<br>314×22⁴⁹⁷+21<br>22⁴⁸⁵−129<br>21×22⁴⁵⁷+5459||http://factordb.com/index.php?id=1100000003594696838&open=prime<br>http://factordb.com/index.php?id=1100000003591359839&open=prime<br>http://factordb.com/index.php?id=1100000003591360774&open=prime<br>http://factordb.com/index.php?id=1100000003591361817&open=prime<br>http://factordb.com/index.php?id=1100000003591362567&open=prime<br>http://factordb.com/index.php?id=1100000000632724415&open=prime<br>http://factordb.com/index.php?id=1100000000632724334&open=prime<br>http://factordb.com/index.php?id=1100000000632703239&open=prime<br>http://factordb.com/index.php?id=1100000003591364730&open=prime<br>http://factordb.com/index.php?id=1100000003591365331&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003594696838&base=22<br>http://factordb.com/index.php?showid=1100000003591359839&base=22<br>http://factordb.com/index.php?showid=1100000003591360774&base=22<br>http://factordb.com/index.php?showid=1100000003591361817&base=22<br>http://factordb.com/index.php?showid=1100000003591362567&base=22<br>http://factordb.com/index.php?showid=1100000000632724415&base=22<br>http://factordb.com/index.php?showid=1100000000632724334&base=22<br>http://factordb.com/index.php?showid=1100000000632703239&base=22<br>http://factordb.com/index.php?showid=1100000003591364730&base=22<br>http://factordb.com/index.php?showid=1100000003591365331&base=22<nowiki/>||0||–|| |- ||23||65178~65265||B0₉₃₀₄₆FB<br>L₈₆₄₄₄D<br>AJ₈₁₀₆₅4<br>20₇₃₅₆₀98<br>J₆₈₂₁₇G4<br>D70₆₆₇₇₀B<br>5F₆₂₃₄₀6<br>A7M7₆₁₅₃₂<br>B30₆₁₁₃₆5<br>EJ₅₂₁₆₉||93049<br>86445<br>81067<br>73563<br>68219<br>66773<br>62342<br>61535<br>61139<br>52170||126708<br>117715<br>110391<br>100172<br>92896<br>90927<br>84893<br>83794<br>83255<br>71042||11×23⁹³⁰⁴⁸+356<br>(21×23⁸⁶⁴⁴⁵−197)/22<br>(239×23⁸¹⁰⁶⁶−349)/22<br>2×23⁷³⁵⁶²+215<br>(19×23⁶⁸²¹⁹−1867)/22<br>306×23⁶⁶⁷⁷¹+11<br>(125×23⁶²³⁴¹−213)/22<br>(120413×23⁶¹⁵³²−7)/22<br>256×23⁶¹¹³⁷+5<br>(327×23⁵²¹⁶⁹−19)/22||http://factordb.com/index.php?id=1100000004691540361&open=prime<br>http://factordb.com/index.php?id=1100000004691546739&open=prime<br>http://factordb.com/index.php?id=1100000004691548070&open=prime<br>http://factordb.com/index.php?id=1100000004691548569&open=prime<br>http://factordb.com/index.php?id=1100000004691549462&open=prime<br>http://factordb.com/index.php?id=1100000004691549803&open=prime<br>http://factordb.com/index.php?id=1100000004691551005&open=prime<br>http://factordb.com/index.php?id=1100000004691556967&open=prime<br>http://factordb.com/index.php?id=1100000004691557254&open=prime<br>http://factordb.com/index.php?id=1100000004691557548&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004691540361&base=23<br>http://factordb.com/index.php?showid=1100000004691546739&base=23<br>http://factordb.com/index.php?showid=1100000004691548070&base=23<br>http://factordb.com/index.php?showid=1100000004691548569&base=23<br>http://factordb.com/index.php?showid=1100000004691549462&base=23<br>http://factordb.com/index.php?showid=1100000004691549803&base=23<br>http://factordb.com/index.php?showid=1100000004691551005&base=23<br>http://factordb.com/index.php?showid=1100000004691556967&base=23<br>http://factordb.com/index.php?showid=1100000004691557254&base=23<br>http://factordb.com/index.php?showid=1100000004691557548&base=23<nowiki/>||87||100000|| |- ||24||3409||N00N₈₁₂₉LN<br>88N₅₉₅₁<br>A0₂₉₅₁8ID<br>D₂₆₉₈LD<br>N₂₆₄₄LLN<br>BC0₃₃₁B<br>20₃₁₃7<br>C7₂₉₈<br>D0₂₅₉KKD<br>I0₂₄₁I5||8134<br>5953<br>2955<br>2700<br>2647<br>334<br>315<br>299<br>263<br>244||11227<br>8216<br>4079<br>3727<br>3654<br>461<br>434<br>413<br>363<br>337||13249×24⁸¹³¹−49<br>201×24⁵⁹⁵¹−1<br>10×24²⁹⁵⁴+5053<br>(13×24²⁷⁰⁰+4403)/23<br>24²⁶⁴⁷−1201<br>276×24³³²+11<br>2×24³¹⁴+7<br>(283×24²⁹⁸−7)/23<br>13×24²⁶²+12013<br>18×24²⁴³+437||http://factordb.com/index.php?id=1100000003593391606&open=prime<br>http://factordb.com/index.php?id=1100000003593275880&open=prime<br>http://factordb.com/index.php?id=1100000003593269654&open=prime<br>http://factordb.com/index.php?id=1100000003593269876&open=prime<br>http://factordb.com/index.php?id=1100000003593270089&open=prime<br>http://factordb.com/index.php?id=1100000002633359842&open=prime<br>http://factordb.com/index.php?id=1100000002355610241&open=prime<br>http://factordb.com/index.php?id=1100000002326181235&open=prime<br>http://factordb.com/index.php?id=1100000003593270725&open=prime<br>http://factordb.com/index.php?id=1100000002633360037&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003593391606&base=24<br>http://factordb.com/index.php?showid=1100000003593275880&base=24<br>http://factordb.com/index.php?showid=1100000003593269654&base=24<br>http://factordb.com/index.php?showid=1100000003593269876&base=24<br>http://factordb.com/index.php?showid=1100000003593270089&base=24<br>http://factordb.com/index.php?showid=1100000002633359842&base=24<br>http://factordb.com/index.php?showid=1100000002355610241&base=24<br>http://factordb.com/index.php?showid=1100000002326181235&base=24<br>http://factordb.com/index.php?showid=1100000003593270725&base=24<br>http://factordb.com/index.php?showid=1100000002633360037&base=24<nowiki/>||0||–|| |- ||25||133639~133724||E₉₈₃₉₆FOO<br>1J710₉₆₂₇₂1<br>NB0₈₅₅₉₈5NH<br>D70₈₁₅₈₁JJ7<br>F0₈₀₀₅₄HL<br>J010₇₅₉₄₃E7<br>K₆₇₇₇₁5I<br>LO₆₆₃₇₇KC<br>KJD0₆₃₃₉₉1<br>70₆₀₈₉₂D711||98399<br>96277<br>85603<br>81586<br>80057<br>75948<br>67773<br>66380<br>63403<br>60897||137556<br>134589<br>119668<br>114053<br>111915<br>106171<br>94743<br>92796<br>88634<br>85130||(7×25⁹⁸³⁹⁹+10613)/12<br>27676×25⁹⁶²⁷³+1<br>586×25⁸⁵⁶⁰¹+3717<br>332×25⁸¹⁵⁸⁴+12357<br>15×25⁸⁰⁰⁵⁶+446<br>11876×25⁷⁵⁹⁴⁵+357<br>(5×25⁶⁷⁷⁷³−2267)/6<br>22×25⁶⁶³⁷⁹−113<br>12988×25⁶³⁴⁰⁰+1<br>7×25⁶⁰⁸⁹⁶+207526||http://factordb.com/index.php?id=1100000000808118215&open=prime<br>http://factordb.com/index.php?id=1100000003983674902&open=prime<br>http://factordb.com/index.php?id=1100000004909706420&open=prime<br>http://factordb.com/index.php?id=1100000004909733266&open=prime<br>http://factordb.com/index.php?id=1100000004909750102&open=prime<br>http://factordb.com/index.php?id=1100000004909770736&open=prime<br>http://factordb.com/index.php?id=1100000004586986394&open=prime<br>http://factordb.com/index.php?id=1100000000808118270&open=prime<br>http://factordb.com/index.php?id=1100000004586986664&open=prime<br>http://factordb.com/index.php?id=1100000004586986798&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118215&base=25<br>http://factordb.com/index.php?showid=1100000003983674902&base=25<br>http://factordb.com/index.php?showid=1100000004909706420&base=25<br>http://factordb.com/index.php?showid=1100000004909733266&base=25<br>http://factordb.com/index.php?showid=1100000004909750102&base=25<br>http://factordb.com/index.php?showid=1100000004909770736&base=25<br>http://factordb.com/index.php?showid=1100000004586986394&base=25<br>http://factordb.com/index.php?showid=1100000000808118270&base=25<br>http://factordb.com/index.php?showid=1100000004586986664&base=25<br>http://factordb.com/index.php?showid=1100000004586986798&base=25<nowiki/>||85||100000|| |- ||26||25256~25259||85M₁₉₇₀₆₀B<br>M0₆₁₁₈₆2BB<br>J0₄₄₃₀₃KCB<br>6K₂₃₃₀₀5<br>LD0₂₀₉₇₅7<br>7₂₀₂₇₉OL<br>5₁₉₃₉₁6F<br>9GDK₁₅₉₂₀P<br>M₈₇₇₂P<br>K0₄₃₆₄I5||197063<br>61190<br>44307<br>23302<br>20978<br>20281<br>19393<br>15924<br>8773<br>4367||278839<br>86583<br>62694<br>32972<br>29684<br>28697<br>27440<br>22532<br>12414<br>6180||(5347×26¹⁹⁷⁰⁶¹−297)/25<br>22×26⁶¹¹⁸⁹+1649<br>19×26⁴⁴³⁰⁶+13843<br>(34×26²³³⁰¹−79)/5<br>559×26²⁰⁹⁷⁶+7<br>(7×26²⁰²⁸¹+11393)/25<br>(26¹⁹³⁹³+179)/5<br>(32569×26¹⁵⁹²¹+21)/5<br>(22×26⁸⁷⁷³+53)/25<br>20×26⁴³⁶⁶+473||http://factordb.com/index.php?id=1100000008573990023&open=prime<br>http://factordb.com/index.php?id=1100000003968169875&open=prime<br>http://factordb.com/index.php?id=1100000003968156595&open=prime<br>http://factordb.com/index.php?id=1100000003892628745&open=prime<br>http://factordb.com/index.php?id=1100000003892628658&open=prime<br>http://factordb.com/index.php?id=1100000003892628605&open=prime<br>http://factordb.com/index.php?id=1100000003850151202&open=prime<br>http://factordb.com/index.php?id=1100000003850155316&open=prime<br>http://factordb.com/index.php?id=1100000000758011195&open=prime<br>http://factordb.com/index.php?id=1100000002634136508&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008573990023&base=26<br>http://factordb.com/index.php?showid=1100000003968169875&base=26<br>http://factordb.com/index.php?showid=1100000003968156595&base=26<br>http://factordb.com/index.php?showid=1100000003892628745&base=26<br>http://factordb.com/index.php?showid=1100000003892628658&base=26<br>http://factordb.com/index.php?showid=1100000003892628605&base=26<br>http://factordb.com/index.php?showid=1100000003850151202&base=26<br>http://factordb.com/index.php?showid=1100000003850155316&base=26<br>http://factordb.com/index.php?showid=1100000000758011195&base=26<br>http://factordb.com/index.php?showid=1100000002634136508&base=26<nowiki/>||3||200000|| |- ||27||102852~102896||CA0F₈₈₈₈₃A<br>GNN0₇₈₇₉₅N<br>O44L₆₆₀₁₆7<br>NJ0₆₄₃₆₉H<br>ME₄₉₆₄₀9G<br>PH0₄₇₈₉₀1<br>QF₄₇₁₆₅AF5<br>J0₄₀₇₉₁PD<br>510₃₉₁₆₄I07<br>NGN0₃₆₃₂₉N||88887<br>78799<br>66020<br>64372<br>49643<br>47893<br>47169<br>40794<br>39169<br>36333||127230<br>112790<br>94499<br>92140<br>71058<br>68553<br>67516<br>58391<br>56065<br>52006||(234483×27⁸⁸⁸⁸⁴−145)/26<br>12308×27⁷⁸⁷⁹⁶+23<br>(457829×27⁶⁶⁰¹⁷−385)/26<br>640×27⁶⁴³⁷⁰+17<br>(293×27⁴⁹⁶⁴²−1736)/13<br>692×27⁴⁷⁸⁹¹+1<br>(691×27⁴⁷¹⁶⁸−95045)/26<br>19×27⁴⁰⁷⁹³+688<br>136×27³⁹¹⁶⁷+13129<br>17222×27³⁶³³⁰+23||http://factordb.com/index.php?id=1100000000808118233&open=prime<br>http://factordb.com/index.php?id=1100000004681348398&open=prime<br>http://factordb.com/index.php?id=1100000004374140861&open=prime<br>http://factordb.com/index.php?id=1100000004374138999&open=prime<br>http://factordb.com/index.php?id=1100000000819229859&open=prime<br>http://factordb.com/index.php?id=1100000004102754118&open=prime<br>http://factordb.com/index.php?id=1100000004102755880&open=prime<br>http://factordb.com/index.php?id=1100000004102758254&open=prime<br>http://factordb.com/index.php?id=1100000004102875088&open=prime<br>http://factordb.com/index.php?id=1100000004103372866&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118233&base=27<br>http://factordb.com/index.php?showid=1100000004681348398&base=27<br>http://factordb.com/index.php?showid=1100000004374140861&base=27<br>http://factordb.com/index.php?showid=1100000004374138999&base=27<br>http://factordb.com/index.php?showid=1100000000819229859&base=27<br>http://factordb.com/index.php?showid=1100000004102754118&base=27<br>http://factordb.com/index.php?showid=1100000004102755880&base=27<br>http://factordb.com/index.php?showid=1100000004102758254&base=27<br>http://factordb.com/index.php?showid=1100000004102875088&base=27<br>http://factordb.com/index.php?showid=1100000004103372866&base=27<nowiki/>||44||100000|| |- ||28||25528~25529||O4O₉₄₅₃₅9<br>5OA₃₁₂₃₈F<br>N6₂₄₀₅₁LR<br>D0₅₂₆₇77D<br>QO₄₂₃₉69<br>5₃₇₄₆8P<br>G0₁₈₉₉AN<br>A₁₄₂₃6F<br>5I₁₃₇₀F<br>5₁₃₃₂P8P||94538<br>31241<br>24054<br>5271<br>4242<br>3748<br>1902<br>1425<br>1372<br>1335||136812<br>45210<br>34810<br>7628<br>6139<br>5424<br>2753<br>2062<br>1985<br>1932||(6092×28⁹⁴⁵³⁶−143)/9<br>(4438×28³¹²³⁹+125)/27<br>(209×28²⁴⁰⁵³+3967)/9<br>13×28⁵²⁷⁰+5697<br>(242×28⁴²⁴¹−4679)/9<br>(5×28³⁷⁴⁸+2803)/27<br>16×28¹⁹⁰¹+303<br>(10×28¹⁴²⁵−2899)/27<br>(17×28¹³⁷¹−11)/3<br>(5×28¹³³⁵+426163)/27||http://factordb.com/index.php?id=1100000000808118231&open=prime<br>http://factordb.com/index.php?id=1100000003880455200&open=prime<br>http://factordb.com/index.php?id=1100000003879667576&open=prime<br>http://factordb.com/index.php?id=1100000003850151420&open=prime<br>http://factordb.com/index.php?id=1100000000840839934&open=prime<br>http://factordb.com/index.php?id=1100000003850161974&open=prime<br>http://factordb.com/index.php?id=1100000003850161973&open=prime<br>http://factordb.com/index.php?id=1100000000840839947&open=prime<br>http://factordb.com/index.php?id=1100000003850161972&open=prime<br>http://factordb.com/index.php?id=1100000003850161965&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118231&base=28<br>http://factordb.com/index.php?showid=1100000003880455200&base=28<br>http://factordb.com/index.php?showid=1100000003879667576&base=28<br>http://factordb.com/index.php?showid=1100000003850151420&base=28<br>http://factordb.com/index.php?showid=1100000000840839934&base=28<br>http://factordb.com/index.php?showid=1100000003850161974&base=28<br>http://factordb.com/index.php?showid=1100000003850161973&base=28<br>http://factordb.com/index.php?showid=1100000000840839947&base=28<br>http://factordb.com/index.php?showid=1100000003850161972&base=28<br>http://factordb.com/index.php?showid=1100000003850161965&base=28<nowiki/>||1||900000|| |- ||29||355242~355367||830₉₉₃₇₇4<br>GP5J₉₄₉₃₅<br>P05J₉₀₂₈₉<br>BBD0₈₈₈₈₈PB<br>8B₈₅₃₃₃G<br>L0₈₁₅₇₁5955<br>E0₇₇₃₇₂L7B<br>LPC₇₅₁₅₁9<br>JR0₇₄₆₂₂7<br>B₇₄₅₀₁0RP||99380<br>94938<br>90292<br>88893<br>85335<br>81576<br>77376<br>75154<br>74625<br>74504||145333<br>138837<br>132043<br>129997<br>124794<br>119297<br>113155<br>109905<br>109132<br>108955||235×29⁹⁹³⁷⁸+4<br>(397227×29⁹⁴⁹³⁵−19)/28<br>(588859×29⁹⁰²⁸⁹−19)/28<br>9583×29⁸⁸⁸⁹⁰+736<br>(235×29⁸⁵³³⁴+129)/14<br>21×29⁸¹⁵⁷⁵+129664<br>14×29⁷⁷³⁷⁵+17875<br>(4441×29⁷⁵¹⁵²−24)/7<br>578×29⁷⁴⁶²³+7<br>(11×29⁷⁴⁵⁰⁴−245655)/28||http://factordb.com/index.php?id=1100000008253882372&open=prime<br>http://factordb.com/index.php?id=1100000008253893542&open=prime<br>http://factordb.com/index.php?id=1100000008253899083&open=prime<br>http://factordb.com/index.php?id=1100000008253909183&open=prime<br>http://factordb.com/index.php?id=1100000008253921388&open=prime<br>http://factordb.com/index.php?id=1100000008253925955&open=prime<br>http://factordb.com/index.php?id=1100000008253931446&open=prime<br>http://factordb.com/index.php?id=1100000000808118236&open=prime<br>http://factordb.com/index.php?id=1100000008253934219&open=prime<br>http://factordb.com/index.php?id=1100000008253936120&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008253882372&base=29<br>http://factordb.com/index.php?showid=1100000008253893542&base=29<br>http://factordb.com/index.php?showid=1100000008253899083&base=29<br>http://factordb.com/index.php?showid=1100000008253909183&base=29<br>http://factordb.com/index.php?showid=1100000008253921388&base=29<br>http://factordb.com/index.php?showid=1100000008253925955&base=29<br>http://factordb.com/index.php?showid=1100000008253931446&base=29<br>http://factordb.com/index.php?showid=1100000000808118236&base=29<br>http://factordb.com/index.php?showid=1100000008253934219&base=29<br>http://factordb.com/index.php?showid=1100000008253936120&base=29<nowiki/>||125||100000|| |- ||30||2619||OT₃₄₂₀₅<br>I0₂₄₆₀₈D<br>5₄₈₈₂J<br>C0₁₀₂₂1<br>M0₅₄₇SS7<br>M₂₄₁QB<br>AN₂₀₆<br>50₁₆₄B<br>J₁₅₃QJ<br>J₉₄QQJ||34206<br>24610<br>4883<br>1024<br>551<br>243<br>207<br>166<br>155<br>97||50527<br>36352<br>7213<br>1513<br>814<br>359<br>306<br>245<br>229<br>144||25×30³⁴²⁰⁵−1<br>18×30²⁴⁶⁰⁹+13<br>(5×30⁴⁸⁸³+401)/29<br>12×30¹⁰²³+1<br>22×30⁵⁵⁰+26047<br>(22×30²⁴³+3139)/29<br>(313×30²⁰⁶−23)/29<br>5×30¹⁶⁵+11<br>(19×30¹⁵⁵+6071)/29<br>(19×30⁹⁷+188771)/29||http://factordb.com/index.php?id=1100000000800812865&open=prime<br>http://factordb.com/index.php?id=1100000003593967511&open=prime<br>http://factordb.com/index.php?id=1100000002327649423&open=prime<br>http://factordb.com/index.php?id=1100000000785448736&open=prime<br>http://factordb.com/index.php?id=1100000003593407988&open=prime<br>http://factordb.com/index.php?id=1100000003593408295&open=prime<br>http://factordb.com/index.php?id=1100000002327651073&open=prime<br>http://factordb.com/index.php?id=1100000002356282476&open=ecm<br>http://factordb.com/index.php?id=1100000003593409109&open=ecm<br>http://factordb.com/index.php?id=1100000003593409165&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000800812865&base=30<br>http://factordb.com/index.php?showid=1100000003593967511&base=30<br>http://factordb.com/index.php?showid=1100000002327649423&base=30<br>http://factordb.com/index.php?showid=1100000000785448736&base=30<br>http://factordb.com/index.php?showid=1100000003593407988&base=30<br>http://factordb.com/index.php?showid=1100000003593408295&base=30<br>http://factordb.com/index.php?showid=1100000002327651073&base=30<br>http://factordb.com/index.php?showid=1100000002356282476&base=30<br>http://factordb.com/index.php?showid=1100000003593409109&base=30<br>http://factordb.com/index.php?showid=1100000003593409165&base=30<nowiki/>||0||–|| |- ||31||569323~569400||2IIF₉₁₈₀₅<br>B0₈₈₃₀₉APO9<br>J0T₇₇₅₁₆<br>J090₇₇₁₂₈NNN<br>D₆₉₈₆₁QO<br>9MH0₆₈₆₃₇D<br>J₆₇₁₆₂D<br>N0₆₆₉₇₁32P<br>DDDQ0₆₄₀₈₈TD<br>U₆₃₈₆₁CM3||91808<br>88314<br>77518<br>77134<br>69863<br>68641<br>67163<br>66975<br>64094<br>63864||136918<br>131708<br>115608<br>115035<br>104191<br>102369<br>100165<br>99884<br>95587<br>95245||(4997×31⁹¹⁸⁰⁵−1)/2<br>11×31⁸⁸³¹³+322688<br>(17699×31⁷⁷⁵¹⁶−29)/30<br>18268×31⁷⁷¹³¹+22839<br>(13×31⁶⁹⁸⁶³+12407)/30<br>9348×31⁶⁸⁶³⁸+13<br>(19×31⁶⁷¹⁶³−199)/30<br>23×31⁶⁶⁹⁷⁴+2970<br>400205×31⁶⁴⁰⁹⁰+912<br>31⁶³⁸⁶⁴−17574||http://factordb.com/index.php?id=1100000007050395732&open=prime<br>http://factordb.com/index.php?id=1100000007050397309&open=prime<br>http://factordb.com/index.php?id=1100000007050398940&open=prime<br>http://factordb.com/index.php?id=1100000007050400178&open=prime<br>http://factordb.com/index.php?id=1100000006965878559&open=prime<br>http://factordb.com/index.php?id=1100000006965875678&open=prime<br>http://factordb.com/index.php?id=1100000006965873668&open=prime<br>http://factordb.com/index.php?id=1100000006965870538&open=prime<br>http://factordb.com/index.php?id=1100000006965868103&open=prime<br>http://factordb.com/index.php?id=1100000006965865343&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000007050395732&base=31<br>http://factordb.com/index.php?showid=1100000007050397309&base=31<br>http://factordb.com/index.php?showid=1100000007050398940&base=31<br>http://factordb.com/index.php?showid=1100000007050400178&base=31<br>http://factordb.com/index.php?showid=1100000006965878559&base=31<br>http://factordb.com/index.php?showid=1100000006965875678&base=31<br>http://factordb.com/index.php?showid=1100000006965873668&base=31<br>http://factordb.com/index.php?showid=1100000006965870538&base=31<br>http://factordb.com/index.php?showid=1100000006965868103&base=31<br>http://factordb.com/index.php?showid=1100000006965865343&base=31<nowiki/>||77||100000|| |- ||32||168882~169002||V₉₉₅₈₃63<br>6₈₉₀₇₄AF<br>8₇₇₇₀₀H<br>Q₇₇₄₀₁EQQQ3<br>8₇₇₂₄₉3<br>JM₇₆₀₂₈L<br>E₇₂₉₁₉IL<br>B0₆₇₆₈₀CB<br>GK₆₆₀₇₆F<br>KN₆₅₀₂₂||99585<br>89076<br>77701<br>77406<br>77250<br>76030<br>72921<br>67683<br>66078<br>65023||149891<br>134073<br>116952<br>116508<br>116273<br>114437<br>109757<br>101873<br>99458<br>97870||32⁹⁹⁵⁸⁵−829<br>(6×32⁸⁹⁰⁷⁶+4241)/31<br>(8×32⁷⁷⁷⁰¹+271)/31<br>(26×32⁷⁷⁴⁰⁶−390071011)/31<br>(8×32⁷⁷²⁵⁰−163)/31<br>(611×32⁷⁶⁰²⁹−53)/31<br>(14×32⁷²⁹²¹+4171)/31<br>11×32⁶⁷⁶⁸²+395<br>(516×32⁶⁶⁰⁷⁷−175)/31<br>(643×32⁶⁵⁰²²−23)/31||http://factordb.com/index.php?id=1100000005514892191&open=prime<br>http://factordb.com/index.php?id=1100000005514897129&open=prime<br>http://factordb.com/index.php?id=1100000005514901700&open=prime<br>http://factordb.com/index.php?id=1100000005514915338&open=prime<br>http://factordb.com/index.php?id=1100000005514918574&open=prime<br>http://factordb.com/index.php?id=1100000005514922523&open=prime<br>http://factordb.com/index.php?id=1100000004591654373&open=prime<br>http://factordb.com/index.php?id=1100000004591654467&open=prime<br>http://factordb.com/index.php?id=1100000004591654632&open=prime<br>http://factordb.com/index.php?id=1100000004591654952&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005514892191&base=32<br>http://factordb.com/index.php?showid=1100000005514897129&base=32<br>http://factordb.com/index.php?showid=1100000005514901700&base=32<br>http://factordb.com/index.php?showid=1100000005514915338&base=32<br>http://factordb.com/index.php?showid=1100000005514918574&base=32<br>http://factordb.com/index.php?showid=1100000005514922523&base=32<br>http://factordb.com/index.php?showid=1100000004591654373&base=32<br>http://factordb.com/index.php?showid=1100000004591654467&base=32<br>http://factordb.com/index.php?showid=1100000004591654632&base=32<br>http://factordb.com/index.php?showid=1100000004591654952&base=32<nowiki/>||120||100000|| |- ||33||280012~280093||DP₉₅₀₉₃M5<br>HJ0₉₄₂₉₅J<br>90₉₃₅₉₇Q<br>9F0₉₃₁₅₇N<br>7₈₉₄₄₉333H<br>K3₈₀₇₅₁6K<br>D₈₀₁₀₇9UD<br>VFU₇₂₂₀₄FK<br>J₆₈₇₁₅2BJ<br>DF0₆₈₃₆₇J||95096<br>94298<br>93599<br>93160<br>89453<br>80754<br>80110<br>72208<br>68718<br>68370||144405<br>143193<br>142131<br>141465<br>135835<br>122626<br>121648<br>109649<br>104350<br>103821||(441×33⁹⁵⁰⁹⁵−3833)/32<br>580×33⁹⁴²⁹⁶+19<br>9×33⁹³⁵⁹⁸+26<br>312×33⁹³¹⁵⁸+23<br>(7×33⁸⁹⁴⁵³−4743239)/32<br>(643×33⁸⁰⁷⁵³+3709)/32<br>(13×33⁸⁰¹¹⁰−121453)/32<br>(16623×33⁷²²⁰⁶−8095)/16<br>(19×33⁶⁸⁷¹⁸−600883)/32<br>444×33⁶⁸³⁶⁸+19||http://factordb.com/index.php?id=1100000005652348775&open=prime<br>http://factordb.com/index.php?id=1100000005652362811&open=prime<br>http://factordb.com/index.php?id=1100000005652375073&open=prime<br>http://factordb.com/index.php?id=1100000005652389776&open=prime<br>http://factordb.com/index.php?id=1100000005652430746&open=prime<br>http://factordb.com/index.php?id=1100000005652446200&open=prime<br>http://factordb.com/index.php?id=1100000005652461592&open=prime<br>http://factordb.com/index.php?id=1100000004614764298&open=prime<br>http://factordb.com/index.php?id=1100000004614770536&open=prime<br>http://factordb.com/index.php?id=1100000004614784274&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005652348775&base=33<br>http://factordb.com/index.php?showid=1100000005652362811&base=33<br>http://factordb.com/index.php?showid=1100000005652375073&base=33<br>http://factordb.com/index.php?showid=1100000005652389776&base=33<br>http://factordb.com/index.php?showid=1100000005652430746&base=33<br>http://factordb.com/index.php?showid=1100000005652446200&base=33<br>http://factordb.com/index.php?showid=1100000005652461592&base=33<br>http://factordb.com/index.php?showid=1100000004614764298&base=33<br>http://factordb.com/index.php?showid=1100000004614770536&base=33<br>http://factordb.com/index.php?showid=1100000004614784274&base=33<nowiki/>||81||100000|| |- ||34||184785~184832||GFGC₉₉₉₉₆5<br>90₉₇₉₅₀FJ<br>NM0₈₅₂₁₈KX<br>F₈₃₁₈₉H2HP<br>P₇₉₄₄₁444P<br>6₇₇₀₂₇8X<br>XQIQ₇₂₂₄₁D<br>T₆₆₅₃₀IF<br>4₆₆₁₅₂B<br>2EEC₆₆₀₃₉7||100000<br>97953<br>85222<br>83193<br>79445<br>77029<br>72245<br>66532<br>66153<br>66043||153148<br>150013<br>130516<br>127408<br>121669<br>117968<br>110642<br>101893<br>101312<br>101143||(209246×34⁹⁹⁹⁹⁷−81)/11<br>9×34⁹⁷⁹⁵²+529<br>804×34⁸⁵²²⁰+713<br>(5×34⁸³¹⁹³+700233)/11<br>(25×34⁷⁹⁴⁴⁵−28062367)/33<br>(2×34⁷⁷⁰²⁹+1043)/11<br>(1288676×34⁷²²⁴²−455)/33<br>(29×34⁶⁶⁵³²−12833)/33<br>(4×34⁶⁶¹⁵³+227)/33<br>(30826×34⁶⁶⁰⁴⁰−59)/11||http://factordb.com/index.php?id=1100000004702891268&open=prime<br>http://factordb.com/index.php?id=1100000004702894713&open=prime<br>http://factordb.com/index.php?id=1100000004702900996&open=prime<br>http://factordb.com/index.php?id=1100000004702910376&open=prime<br>http://factordb.com/index.php?id=1100000004702913746&open=prime<br>http://factordb.com/index.php?id=1100000004702918600&open=prime<br>http://factordb.com/index.php?id=1100000004399656529&open=prime<br>http://factordb.com/index.php?id=1100000004399657696&open=prime<br>http://factordb.com/index.php?id=1100000004399658651&open=prime<br>http://factordb.com/index.php?id=1100000004399659716&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004702891268&base=34<br>http://factordb.com/index.php?showid=1100000004702894713&base=34<br>http://factordb.com/index.php?showid=1100000004702900996&base=34<br>http://factordb.com/index.php?showid=1100000004702910376&base=34<br>http://factordb.com/index.php?showid=1100000004702913746&base=34<br>http://factordb.com/index.php?showid=1100000004702918600&base=34<br>http://factordb.com/index.php?showid=1100000004399656529&base=34<br>http://factordb.com/index.php?showid=1100000004399657696&base=34<br>http://factordb.com/index.php?showid=1100000004399658651&base=34<br>http://factordb.com/index.php?showid=1100000004399659716&base=34<nowiki/>||47||100000|| |- ||35||720002~720062||N0N₉₉₉₇₁9<br>V0₈₃₆₆₉E73<br>N₈₁₅₆₃K7N<br>BJ0₈₁₂₇₉N<br>J0₈₀₀₆₂FUH<br>43V₇₉₇₅₄<br>9₇₆₆₀₀K3<br>LB₇₁₃₆₆PB<br>Q₆₄₁₅₀H<br>50₆₃₃₉₇5R||99974<br>83673<br>81566<br>81282<br>80066<br>79756<br>76602<br>71369<br>64151<br>63400||154367<br>129197<br>125944<br>125505<br>123628<br>123148<br>118279<br>110199<br>99054<br>97894||(27393×35⁹⁹⁹⁷²−499)/34<br>31×35⁸³⁶⁷²+17398<br>(23×35⁸¹⁵⁶⁶−144013)/34<br>404×35⁸¹²⁸⁰+23<br>19×35⁸⁰⁰⁶⁵+19442<br>(4893×35⁷⁹⁷⁵⁴−31)/34<br>(9×35⁷⁶⁶⁰²+12877)/34<br>(725×35⁷¹³⁶⁸+16649)/34<br>(13×35⁶⁴¹⁵¹−166)/17<br>5×35⁶³³⁹⁹+202||http://factordb.com/index.php?id=1100000008248342445&open=prime<br>http://factordb.com/index.php?id=1100000008248353306&open=prime<br>http://factordb.com/index.php?id=1100000008248375642&open=prime<br>http://factordb.com/index.php?id=1100000008248397018&open=prime<br>http://factordb.com/index.php?id=1100000008248412468&open=prime<br>http://factordb.com/index.php?id=1100000008248418540&open=prime<br>http://factordb.com/index.php?id=1100000008248423670&open=prime<br>http://factordb.com/index.php?id=1100000008192119974&open=prime<br>http://factordb.com/index.php?id=1100000008192126630&open=prime<br>http://factordb.com/index.php?id=1100000008192129294&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008248342445&base=35<br>http://factordb.com/index.php?showid=1100000008248353306&base=35<br>http://factordb.com/index.php?showid=1100000008248375642&base=35<br>http://factordb.com/index.php?showid=1100000008248397018&base=35<br>http://factordb.com/index.php?showid=1100000008248412468&base=35<br>http://factordb.com/index.php?showid=1100000008248418540&base=35<br>http://factordb.com/index.php?showid=1100000008248423670&base=35<br>http://factordb.com/index.php?showid=1100000008192119974&base=35<br>http://factordb.com/index.php?showid=1100000008192126630&base=35<br>http://factordb.com/index.php?showid=1100000008192129294&base=35<nowiki/>||60||100000|| |- ||36||35286~35290||P₈₁₉₉₃SZ<br>S0₇₅₀₀₇8H<br>7K₂₆₅₆₇Z<br>J₁₀₁₁₇LJ<br>VL0₇₂₅₈J<br>EO0₆₁₇₇V<br>FZ₅₇₇₇3P<br>T09₄₆₁₈1<br>RY₄₅₆₂H<br>OZ₃₉₃₂AZ||81995<br>75010<br>26569<br>10119<br>7261<br>6180<br>5780<br>4621<br>4564<br>3935||127609<br>116739<br>41349<br>15748<br>11301<br>9618<br>8996<br>7192<br>7103<br>6124||(5×36⁸¹⁹⁹⁵+821)/7<br>28×36⁷⁵⁰⁰⁹+305<br>(53×36²⁶⁵⁶⁸+101)/7<br>(19×36¹⁰¹¹⁹+2501)/35<br>1137×36⁷²⁵⁹+19<br>528×36⁶¹⁷⁸+31<br>16×36⁵⁷⁷⁹−1163<br>(36549×36⁴⁶¹⁹−289)/35<br>(979×36⁴⁵⁶³−629)/35<br>25×36³⁹³⁴−901||http://factordb.com/index.php?id=1100000002394962083&open=prime<br>http://factordb.com/index.php?id=1100000004020085177&open=prime<br>http://factordb.com/index.php?id=1100000003896952461&open=prime<br>http://factordb.com/index.php?id=1100000003807362491&open=prime<br>http://factordb.com/index.php?id=1100000003807362489&open=prime<br>http://factordb.com/index.php?id=1100000003807362488&open=prime<br>http://factordb.com/index.php?id=1100000003807362487&open=prime<br>http://factordb.com/index.php?id=1100000003807362486&open=prime<br>http://factordb.com/index.php?id=1100000003807362485&open=prime<br>http://factordb.com/index.php?id=1100000000840634476&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000002394962083&base=36<br>http://factordb.com/index.php?showid=1100000004020085177&base=36<br>http://factordb.com/index.php?showid=1100000003896952461&base=36<br>http://factordb.com/index.php?showid=1100000003807362491&base=36<br>http://factordb.com/index.php?showid=1100000003807362489&base=36<br>http://factordb.com/index.php?showid=1100000003807362488&base=36<br>http://factordb.com/index.php?showid=1100000003807362487&base=36<br>http://factordb.com/index.php?showid=1100000003807362486&base=36<br>http://factordb.com/index.php?showid=1100000003807362485&base=36<br>http://factordb.com/index.php?showid=1100000000840634476&base=36<nowiki/>||4||200000|| |} == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] 94rzvxfrcttqqh1k9mquyqznkwcin7q 2819706 2819705 2026-07-28T19:40:54Z Athene241 3100061 /* Solve the problem */ 2819706 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 11 family 2{5}: Always divisible by 2 or 3 * The base 14 family B{0}1: Always divisible by 3 or 5 * The base 13 family 95{0}3: Always divisible by 5, 7, or 17 * The base 16 family {4}D: Always divisible by 3, 7, or 13 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family {1}: Can be written as (9^''n''−1)/8 and can be factored as (3^''n''−1) × (3^''n''+1) / 8 * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 16 family 1{5}: Can be written as (4×16^''n''−1)/3 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) / 3 * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 27 family 7{Q}: Can be written as 8×27^''n''−1 and can be factored as (2×3^''n''−1) × (4×9^''n''+2×3^''n''+1) * The base 27 family 9{G}: Can be written as (125×27^''n''−8)/13 and can be factored as (5×3^''n''−2) × (25×9^''n''+10×3^''n''+4) * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 14 family {D}5: Can be written as 14^''n''−9, it is divisible by 5 if ''n'' is odd and can be factored as (14^''n''/2−3) × (14^''n''/2+3) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 17 family 7{9}: Can be written as (121×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17^''n''/2−3) × (11×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even * The base 19 family 7{2}: Can be written as (64×19^''n''−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19^''n''/2−1) × (8×19^''n''/2+1) / 9 if ''n'' is even * The base 24 family 3{N}: Can be written as 4×24^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24^''n''/2−1) × (2×24^''n''/2+1) if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored). The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]]. To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''^''n''+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1 we only used it to sieve the sequence ''a''×''b''^''n''+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''^''n''+''c'', but 2 may not divide (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the "easy" bases (bases ''b'' with ≤ 150 primes > 10²⁹⁹ (base ''b'' = 26 has 83 known primes > 10²⁹⁹ and 3 unsolved families, base ''b'' = 36 has 75 known primes > 10²⁹⁹ and 4 unsolved families, base ''b'' = 17 has 99 known primes > 10²⁹⁹ and 18 unsolved families, base ''b'' = 21 has 80 known primes > 10²⁹⁹ and 12 unsolved families, base ''b'' = 19 has 201 known primes > 10²⁹⁹ and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation. == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == Condensed table for bases 2 ≤ ''b'' ≤ 36 == {|class="wikitable" ||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write "''d''_''n''" if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)|| |- ||2||1||11||2||1||3||http://factordb.com/index.php?id=3&open=ecm||http://factordb.com/index.php?showid=3&base=2||0||–|| |- ||3||3||111<br>21<br>12||3<br>2<br>2||2<br>1<br>1||13<br>7<br>5||http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=13&base=3<br>http://factordb.com/index.php?showid=7&base=3<br>http://factordb.com/index.php?showid=5&base=3<nowiki/>||0||–|| |- ||4||5||221<br>31<br>23<br>13<br>11||3<br>2<br>2<br>2<br>2||2<br>2<br>2<br>1<br>1||41<br>13<br>11<br>7<br>5||http://factordb.com/index.php?id=41&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=41&base=4<br>http://factordb.com/index.php?showid=13&base=4<br>http://factordb.com/index.php?showid=11&base=4<br>http://factordb.com/index.php?showid=7&base=4<br>http://factordb.com/index.php?showid=5&base=4<nowiki/>||0||–|| |- ||5||22||10₉₃13<br>300031<br>44441<br>33331<br>33001<br>30301<br>14444<br>10103<br>3101<br>414||96<br>6<br>5<br>5<br>5<br>5<br>5<br>5<br>4<br>3||67<br>4<br>4<br>4<br>4<br>4<br>4<br>3<br>3<br>3||5⁹⁵+8<br>9391<br>3121<br>2341<br>2251<br>1951<br>1249<br>653<br>401<br>109||http://factordb.com/index.php?id=1100000000034686071&open=ecm<br>http://factordb.com/index.php?id=9391&open=ecm<br>http://factordb.com/index.php?id=3121&open=ecm<br>http://factordb.com/index.php?id=2341&open=ecm<br>http://factordb.com/index.php?id=2251&open=ecm<br>http://factordb.com/index.php?id=1951&open=ecm<br>http://factordb.com/index.php?id=1249&open=ecm<br>http://factordb.com/index.php?id=653&open=ecm<br>http://factordb.com/index.php?id=401&open=ecm<br>http://factordb.com/index.php?id=109&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000034686071&base=5<br>http://factordb.com/index.php?showid=9391&base=5<br>http://factordb.com/index.php?showid=3121&base=5<br>http://factordb.com/index.php?showid=2341&base=5<br>http://factordb.com/index.php?showid=2251&base=5<br>http://factordb.com/index.php?showid=1951&base=5<br>http://factordb.com/index.php?showid=1249&base=5<br>http://factordb.com/index.php?showid=653&base=5<br>http://factordb.com/index.php?showid=401&base=5<br>http://factordb.com/index.php?showid=109&base=5<nowiki/>||0||–|| |- ||6||11||40041<br>4441<br>4401<br>51<br>45<br>35<br>31<br>25<br>21<br>15||5<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||4<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||5209<br>1033<br>1009<br>31<br>29<br>23<br>19<br>17<br>13<br>11||http://factordb.com/index.php?id=5209&open=ecm<br>http://factordb.com/index.php?id=1033&open=ecm<br>http://factordb.com/index.php?id=1009&open=ecm<br>http://factordb.com/index.php?id=31&open=ecm<br>http://factordb.com/index.php?id=29&open=ecm<br>http://factordb.com/index.php?id=23&open=ecm<br>http://factordb.com/index.php?id=19&open=ecm<br>http://factordb.com/index.php?id=17&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<nowiki/>||http://factordb.com/index.php?showid=5209&base=6<br>http://factordb.com/index.php?showid=1033&base=6<br>http://factordb.com/index.php?showid=1009&base=6<br>http://factordb.com/index.php?showid=31&base=6<br>http://factordb.com/index.php?showid=29&base=6<br>http://factordb.com/index.php?showid=23&base=6<br>http://factordb.com/index.php?showid=19&base=6<br>http://factordb.com/index.php?showid=17&base=6<br>http://factordb.com/index.php?showid=13&base=6<br>http://factordb.com/index.php?showid=11&base=6<nowiki/>||0||–|| |- ||7||71||3₁₆1<br>510₇1<br>3₆01<br>1100021<br>531101<br>351101<br>300053<br>150001<br>100121<br>40054||17<br>10<br>8<br>7<br>6<br>6<br>6<br>6<br>6<br>5||15<br>9<br>7<br>6<br>5<br>5<br>5<br>5<br>5<br>4||(7¹⁷−5)/2<br>36×7⁸+1<br>(7⁸−47)/2<br>134471<br>91631<br>62819<br>50459<br>28813<br>16871<br>9643||http://factordb.com/index.php?id=116315256993601&open=ecm<br>http://factordb.com/index.php?id=207532837&open=ecm<br>http://factordb.com/index.php?id=2882377&open=ecm<br>http://factordb.com/index.php?id=134471&open=ecm<br>http://factordb.com/index.php?id=91631&open=ecm<br>http://factordb.com/index.php?id=62819&open=ecm<br>http://factordb.com/index.php?id=50459&open=ecm<br>http://factordb.com/index.php?id=28813&open=ecm<br>http://factordb.com/index.php?id=16871&open=ecm<br>http://factordb.com/index.php?id=9643&open=ecm<nowiki/>||http://factordb.com/index.php?showid=116315256993601&base=7<br>http://factordb.com/index.php?showid=207532837&base=7<br>http://factordb.com/index.php?showid=2882377&base=7<br>http://factordb.com/index.php?showid=134471&base=7<br>http://factordb.com/index.php?showid=91631&base=7<br>http://factordb.com/index.php?showid=62819&base=7<br>http://factordb.com/index.php?showid=50459&base=7<br>http://factordb.com/index.php?showid=28813&base=7<br>http://factordb.com/index.php?showid=16871&base=7<br>http://factordb.com/index.php?showid=9643&base=7<nowiki/>||0||–|| |- ||8||75||4₂₂₀7<br>5₁₃25<br>7₁₂1<br>77774₆1<br>74₇1<br>4₈1<br>5₅025<br>5550525<br>5500525<br>4₅77||221<br>15<br>13<br>11<br>9<br>9<br>8<br>7<br>7<br>7||200<br>14<br>12<br>10<br>9<br>8<br>8<br>7<br>7<br>7||(4×8²²¹+17)/7<br>(5×8¹⁵−173)/7<br>8¹³−7<br>(28669×8⁷−25)/7<br>(53×8⁸−25)/7<br>(4×8⁹−25)/7<br>(5×8⁸−2413)/7<br>1495381<br>1474901<br>(4×8⁷+185)/7||http://factordb.com/index.php?id=1100000000416605822&open=ecm<br>http://factordb.com/index.php?id=25131694349141&open=ecm<br>http://factordb.com/index.php?id=549755813881&open=ecm<br>http://factordb.com/index.php?id=8589035809&open=ecm<br>http://factordb.com/index.php?id=127027489&open=ecm<br>http://factordb.com/index.php?id=76695841&open=ecm<br>http://factordb.com/index.php?id=11983381&open=ecm<br>http://factordb.com/index.php?id=1495381&open=ecm<br>http://factordb.com/index.php?id=1474901&open=ecm<br>http://factordb.com/index.php?id=1198399&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000416605822&base=8<br>http://factordb.com/index.php?showid=25131694349141&base=8<br>http://factordb.com/index.php?showid=549755813881&base=8<br>http://factordb.com/index.php?showid=8589035809&base=8<br>http://factordb.com/index.php?showid=127027489&base=8<br>http://factordb.com/index.php?showid=76695841&base=8<br>http://factordb.com/index.php?showid=11983381&base=8<br>http://factordb.com/index.php?showid=1495381&base=8<br>http://factordb.com/index.php?showid=1474901&base=8<br>http://factordb.com/index.php?showid=1198399&base=8<nowiki/>||0||–|| |- ||9||151||30₁₁₅₈11<br>27₆₈₆07<br>76₃₂₉2<br>561₃₆<br>10₂₅57<br>30₂₀51<br>8₁₉335<br>727₁₅07<br>51₁₃61<br>10₁₁507||1161<br>689<br>331<br>38<br>28<br>23<br>22<br>19<br>16<br>15||1108<br>657<br>316<br>37<br>26<br>22<br>21<br>19<br>16<br>14||3×9¹¹⁶⁰+10<br>(23×9⁶⁸⁸−511)/8<br>(31×9³³⁰−19)/4<br>(409×9³⁶−1)/8<br>9²⁷+52<br>3×9²²+46<br>9²²−454<br>(527×9¹⁷−511)/8<br>(41×9¹⁵+359)/8<br>9¹⁴+412||http://factordb.com/index.php?id=1100000002376318423&open=prime<br>http://factordb.com/index.php?id=1100000002495467486&open=prime<br>http://factordb.com/index.php?id=1100000002359003642&open=prime<br>http://factordb.com/index.php?id=1100000001554010824&open=ecm<br>http://factordb.com/index.php?id=1100000002512830927&open=ecm<br>http://factordb.com/index.php?id=1100000000032261811&open=ecm<br>http://factordb.com/index.php?id=1100000002495736583&open=ecm<br>http://factordb.com/index.php?id=1100000003446800389&open=ecm<br>http://factordb.com/index.php?id=1055192051985121&open=ecm<br>http://factordb.com/index.php?id=22876792455373&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002376318423&base=9<br>http://factordb.com/index.php?showid=1100000002495467486&base=9<br>http://factordb.com/index.php?showid=1100000002359003642&base=9<br>http://factordb.com/index.php?showid=1100000001554010824&base=9<br>http://factordb.com/index.php?showid=1100000002512830927&base=9<br>http://factordb.com/index.php?showid=1100000000032261811&base=9<br>http://factordb.com/index.php?showid=1100000002495736583&base=9<br>http://factordb.com/index.php?showid=1100000003446800389&base=9<br>http://factordb.com/index.php?showid=1055192051985121&base=9<br>http://factordb.com/index.php?showid=22876792455373&base=9<nowiki/>||0||–|| |- ||10||77||50₂₈27<br>5₁₁1<br>805₅1<br>66600049<br>66000049<br>60₅49<br>220₅1<br>5200007<br>946669<br>666649||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||5×10³⁰+27<br>(5×10¹²−41)/9<br>(725×10⁶−41)/9<br>66600049<br>66000049<br>6×10⁷+49<br>22×10⁶+1<br>5200007<br>946669<br>666649||http://factordb.com/index.php?id=1100000000204142046&open=ecm<br>http://factordb.com/index.php?id=555555555551&open=ecm<br>http://factordb.com/index.php?id=80555551&open=ecm<br>http://factordb.com/index.php?id=66600049&open=ecm<br>http://factordb.com/index.php?id=66000049&open=ecm<br>http://factordb.com/index.php?id=60000049&open=ecm<br>http://factordb.com/index.php?id=22000001&open=ecm<br>http://factordb.com/index.php?id=5200007&open=ecm<br>http://factordb.com/index.php?id=946669&open=ecm<br>http://factordb.com/index.php?id=666649&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000204142046&base=10<br>http://factordb.com/index.php?showid=555555555551&base=10<br>http://factordb.com/index.php?showid=80555551&base=10<br>http://factordb.com/index.php?showid=66600049&base=10<br>http://factordb.com/index.php?showid=66000049&base=10<br>http://factordb.com/index.php?showid=60000049&base=10<br>http://factordb.com/index.php?showid=22000001&base=10<br>http://factordb.com/index.php?showid=5200007&base=10<br>http://factordb.com/index.php?showid=946669&base=10<br>http://factordb.com/index.php?showid=666649&base=10<nowiki/>||0||–|| |- ||11||1068||57₆₂₆₆₈<br>557₁₀₁₁<br>7₇₅₉44<br>A₇₁₃58<br>85₂₂₀05<br>507₂₀₆<br>5₁₆₁2A<br>50₁₂₆57<br>10₁₂₅51<br>326₁₂₂||62669<br>1013<br>761<br>715<br>223<br>208<br>163<br>129<br>128<br>124||65263<br>1055<br>793<br>745<br>233<br>217<br>170<br>134<br>133<br>129||(57×11⁶²⁶⁶⁸−7)/10<br>(607×11¹⁰¹¹−7)/10<br>(7×11⁷⁶¹−367)/10<br>11⁷¹⁵−58<br>(17×11²²²−111)/2<br>(557×11²⁰⁶−7)/10<br>(11¹⁶³−57)/2<br>5×11¹²⁸+62<br>11¹²⁷+56<br>(178×11¹²²−3)/5||http://factordb.com/index.php?id=1100000003573679860&open=prime<br>http://factordb.com/index.php?id=1100000002361376522&open=prime<br>http://factordb.com/index.php?id=1100000002505568840&open=prime<br>http://factordb.com/index.php?id=1100000003576826487&open=prime<br>http://factordb.com/index.php?id=1100000003576826769&open=ecm<br>http://factordb.com/index.php?id=1100000002518512744&open=ecm<br>http://factordb.com/index.php?id=1100000002391585327&open=ecm<br>http://factordb.com/index.php?id=1100000002632393378&open=ecm<br>http://factordb.com/index.php?id=1100000002391531300&open=ecm<br>http://factordb.com/index.php?id=1100000003576826781&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003573679860&base=11<br>http://factordb.com/index.php?showid=1100000002361376522&base=11<br>http://factordb.com/index.php?showid=1100000002505568840&base=11<br>http://factordb.com/index.php?showid=1100000003576826487&base=11<br>http://factordb.com/index.php?showid=1100000003576826769&base=11<br>http://factordb.com/index.php?showid=1100000002518512744&base=11<br>http://factordb.com/index.php?showid=1100000002391585327&base=11<br>http://factordb.com/index.php?showid=1100000002632393378&base=11<br>http://factordb.com/index.php?showid=1100000002391531300&base=11<br>http://factordb.com/index.php?showid=1100000003576826781&base=11<nowiki/>||0||–|| |- ||12||106||40₃₉77<br>B0₂₇9B<br>B₆99B<br>AA0₅1<br>B00099B<br>AAA0001<br>BBBAA1<br>A00065<br>44AAA1<br>BBBB1||42<br>30<br>9<br>8<br>7<br>7<br>6<br>6<br>6<br>5||45<br>33<br>10<br>9<br>8<br>8<br>7<br>7<br>7<br>6||4×12⁴¹+91<br>11×12²⁹+119<br>12⁹−313<br>130×12⁶+1<br>32847239<br>32555521<br>2985817<br>2488397<br>1097113<br>248821||http://factordb.com/index.php?id=1100000002375054575&open=ecm<br>http://factordb.com/index.php?id=1100000002354113100&open=ecm<br>http://factordb.com/index.php?id=5159780039&open=ecm<br>http://factordb.com/index.php?id=388177921&open=ecm<br>http://factordb.com/index.php?id=32847239&open=ecm<br>http://factordb.com/index.php?id=32555521&open=ecm<br>http://factordb.com/index.php?id=2985817&open=ecm<br>http://factordb.com/index.php?id=2488397&open=ecm<br>http://factordb.com/index.php?id=1097113&open=ecm<br>http://factordb.com/index.php?id=248821&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002375054575&base=12<br>http://factordb.com/index.php?showid=1100000002354113100&base=12<br>http://factordb.com/index.php?showid=5159780039&base=12<br>http://factordb.com/index.php?showid=388177921&base=12<br>http://factordb.com/index.php?showid=32847239&base=12<br>http://factordb.com/index.php?showid=32555521&base=12<br>http://factordb.com/index.php?showid=2985817&base=12<br>http://factordb.com/index.php?showid=2488397&base=12<br>http://factordb.com/index.php?showid=1097113&base=12<br>http://factordb.com/index.php?showid=248821&base=12<nowiki/>||0||–|| |- ||13||3197||A3₅₉₂₁₉₇A<br>95₁₉₇₄₂₀<br>80₃₂₀₁₇111<br>C5₂₃₇₅₅C<br>C₁₀₆₃₁92<br>B0₆₅₄₀BBA<br>390₆₂₆₆1<br>1770₂₇₀₃17<br>720₂₂₉₇2<br>930₁₅₅₁1||592199<br>197421<br>32021<br>23757<br>10633<br>6544<br>6269<br>2708<br>2300<br>1554||659677<br>219916<br>35670<br>26464<br>11845<br>7290<br>6983<br>3016<br>2562<br>1731||(41×13⁵⁹²¹⁹⁸+27)/4<br>(113×13¹⁹⁷⁴²⁰−5)/12<br>8×13³²⁰²⁰+183<br>(149×13²³⁷⁵⁶+79)/12<br>13¹⁰⁶³³−50<br>11×13⁶⁵⁴³+2012<br>48×13⁶²⁶⁷+1<br>267×13²⁷⁰⁵+20<br>93×13²²⁹⁸+2<br>120×13¹⁵⁵²+1||http://factordb.com/index.php?id=1100000005489162806&open=prime<br>http://factordb.com/index.php?id=1100000003943359311&open=prime<br>http://factordb.com/index.php?id=1100000000490878060&open=prime<br>http://factordb.com/index.php?id=1100000003590647776&open=prime<br>http://factordb.com/index.php?id=1100000003590493750&open=prime<br>http://factordb.com/index.php?id=1100000002616382906&open=prime<br>http://factordb.com/index.php?id=1100000000765961441&open=prime<br>http://factordb.com/index.php?id=1100000003590430825&open=prime<br>http://factordb.com/index.php?id=1100000002632396910&open=prime<br>http://factordb.com/index.php?id=1100000000765961452&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005489162806&base=13<br>http://factordb.com/index.php?showid=1100000003943359311&base=13<br>http://factordb.com/index.php?showid=1100000000490878060&base=13<br>http://factordb.com/index.php?showid=1100000003590647776&base=13<br>http://factordb.com/index.php?showid=1100000003590493750&base=13<br>http://factordb.com/index.php?showid=1100000002616382906&base=13<br>http://factordb.com/index.php?showid=1100000000765961441&base=13<br>http://factordb.com/index.php?showid=1100000003590430825&base=13<br>http://factordb.com/index.php?showid=1100000002632396910&base=13<br>http://factordb.com/index.php?showid=1100000000765961452&base=13<nowiki/>||0||–|| |- ||14||650||4D₁₉₆₉₈<br>34D₇₀₈<br>8D₁₄₁85<br>8₈₆B<br>40₈₃49<br>8C₇₉3<br>18₇₉B<br>6B₇₇2B<br>4₆₃09<br>A₅₉3||19699<br>710<br>144<br>87<br>86<br>81<br>81<br>80<br>65<br>60||22578<br>814<br>165<br>100<br>99<br>93<br>92<br>92<br>74<br>69||5×14¹⁹⁶⁹⁸−1<br>47×14⁷⁰⁸−1<br>9×14¹⁴³−79<br>(8×14⁸⁷+31)/13<br>4×14⁸⁵+65<br>(116×14⁸⁰−129)/13<br>(21×14⁸⁰+31)/13<br>(89×14⁷⁹−1649)/13<br>(4×14⁶⁵−667)/13<br>(10×14⁶⁰−101)/13||http://factordb.com/index.php?id=1100000000884560233&open=prime<br>http://factordb.com/index.php?id=1100000001540144903&open=prime<br>http://factordb.com/index.php?id=1100000003575856650&open=ecm<br>http://factordb.com/index.php?id=1100000002321014379&open=ecm<br>http://factordb.com/index.php?id=1100000000823937973&open=ecm<br>http://factordb.com/index.php?id=1100000002631073246&open=ecm<br>http://factordb.com/index.php?id=1100000002384401372&open=ecm<br>http://factordb.com/index.php?id=1100000002631077787&open=ecm<br>http://factordb.com/index.php?id=1100000000840126683&open=ecm<br>http://factordb.com/index.php?id=1100000002321038522&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000884560233&base=14<br>http://factordb.com/index.php?showid=1100000001540144903&base=14<br>http://factordb.com/index.php?showid=1100000003575856650&base=14<br>http://factordb.com/index.php?showid=1100000002321014379&base=14<br>http://factordb.com/index.php?showid=1100000000823937973&base=14<br>http://factordb.com/index.php?showid=1100000002631073246&base=14<br>http://factordb.com/index.php?showid=1100000002384401372&base=14<br>http://factordb.com/index.php?showid=1100000002631077787&base=14<br>http://factordb.com/index.php?showid=1100000000840126683&base=14<br>http://factordb.com/index.php?showid=1100000002321038522&base=14<nowiki/>||0||–|| |- ||15||1284||7₁₅₅97<br>E₁₄₅397<br>96₁₀₄08<br>7₇₃CE<br>7₅₉CCE<br>50₃₃17<br>EB₃₁<br>6330₂₆1<br>7050₂₄B<br>B70₂₄1||157<br>148<br>107<br>75<br>62<br>36<br>32<br>30<br>28<br>27||185<br>175<br>126<br>88<br>73<br>42<br>38<br>35<br>33<br>32||(15¹⁵⁷+59)/2<br>15¹⁴⁸−2558<br>(66×15¹⁰⁶−619)/7<br>(15⁷⁵+163)/2<br>(15⁶²+2413)/2<br>5×15³⁵+22<br>(207×15³¹−11)/14<br>1398×15²⁷+1<br>1580×15²⁵+11<br>172×15²⁵+1||http://factordb.com/index.php?id=1100000002454891840&open=ecm<br>http://factordb.com/index.php?id=1100000002454900849&open=ecm<br>http://factordb.com/index.php?id=1100000000823937997&open=ecm<br>http://factordb.com/index.php?id=1100000003588407143&open=ecm<br>http://factordb.com/index.php?id=1100000003588407386&open=ecm<br>http://factordb.com/index.php?id=1100000002632398579&open=ecm<br>http://factordb.com/index.php?id=1100000002321033312&open=ecm<br>http://factordb.com/index.php?id=1100000002391199877&open=ecm<br>http://factordb.com/index.php?id=1100000003588407806&open=ecm<br>http://factordb.com/index.php?id=1100000000851967288&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002454891840&base=15<br>http://factordb.com/index.php?showid=1100000002454900849&base=15<br>http://factordb.com/index.php?showid=1100000000823937997&base=15<br>http://factordb.com/index.php?showid=1100000003588407143&base=15<br>http://factordb.com/index.php?showid=1100000003588407386&base=15<br>http://factordb.com/index.php?showid=1100000002632398579&base=15<br>http://factordb.com/index.php?showid=1100000002321033312&base=15<br>http://factordb.com/index.php?showid=1100000002391199877&base=15<br>http://factordb.com/index.php?showid=1100000003588407806&base=15<br>http://factordb.com/index.php?showid=1100000000851967288&base=15<nowiki/>||0||–|| |- ||16||2347||3₁₁₆₁₃₇AF<br>4₇₂₇₈₅DD<br>DB₃₂₂₃₄<br>D0B₁₇₈₀₄<br>5BC₃₇₀₀D<br>90₃₅₄₂91<br>300F₁₉₆₀AF<br>20₁₇₁₃321<br>F8₁₅₁₇F<br>FAF₁₀₆₂45||116139<br>72787<br>32235<br>17806<br>3703<br>3545<br>1965<br>1717<br>1519<br>1066||139845<br>87644<br>38815<br>21441<br>4459<br>4269<br>2366<br>2067<br>1830<br>1284||(16¹¹⁶¹³⁹+619)/5<br>(4×16⁷²⁷⁸⁷+2291)/15<br>(206×16³²²³⁴−11)/15<br>(3131×16¹⁷⁸⁰⁴−11)/15<br>(459×16³⁷⁰¹+1)/5<br>9×16³⁵⁴⁴+145<br>769×16¹⁹⁶²−81<br>2×16¹⁷¹⁶+801<br>(233×16¹⁵¹⁸+97)/15<br>251×16¹⁰⁶⁴−187||http://factordb.com/index.php?id=1100000003851731988&open=prime<br>http://factordb.com/index.php?id=1100000003615909841&open=prime<br>http://factordb.com/index.php?id=1100000002383583629&open=prime<br>http://factordb.com/index.php?id=1100000003589278511&open=prime<br>http://factordb.com/index.php?id=1100000000993764322&open=prime<br>http://factordb.com/index.php?id=1100000000633424191&open=prime<br>http://factordb.com/index.php?id=1100000003588368750&open=prime<br>http://factordb.com/index.php?id=1100000003588386735&open=prime<br>http://factordb.com/index.php?id=1100000000633744824&open=prime<br>http://factordb.com/index.php?id=1100000003588387610&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003851731988&base=16<br>http://factordb.com/index.php?showid=1100000003615909841&base=16<br>http://factordb.com/index.php?showid=1100000002383583629&base=16<br>http://factordb.com/index.php?showid=1100000003589278511&base=16<br>http://factordb.com/index.php?showid=1100000000993764322&base=16<br>http://factordb.com/index.php?showid=1100000000633424191&base=16<br>http://factordb.com/index.php?showid=1100000003588368750&base=16<br>http://factordb.com/index.php?showid=1100000003588386735&base=16<br>http://factordb.com/index.php?showid=1100000000633744824&base=16<br>http://factordb.com/index.php?showid=1100000003588387610&base=16<nowiki/>||0||–|| |- ||17||10415~10427||95F₁₉₈₈₅₅<br>B0₁₈₉₀₈₃DB<br>F70₁₈₆₇₆₇1<br>970₁₆₆₀₄₇1<br>510₁₂₄₀₇₄D<br>49₁₁₁₃₃₃<br>B₆₇₁₀₃2E<br>570₅₁₃₁₀1<br>E9B₄₄₇₃₂<br>D0GD₃₇₀₉₆||198857<br>189086<br>186770<br>166050<br>124077<br>111334<br>67105<br>51313<br>44734<br>37099||244684<br>232661<br>229811<br>204316<br>152670<br>136991<br>82570<br>63138<br>55043<br>45649||(2543×17¹⁹⁸⁸⁵⁵−15)/16<br>11×17¹⁸⁹⁰⁸⁵+232<br>262×17¹⁸⁶⁷⁶⁸+1<br>160×17¹⁶⁶⁰⁴⁸+1<br>86×17¹²⁴⁰⁷⁵+13<br>(73×17¹¹¹³³³−9)/16<br>(11×17⁶⁷¹⁰⁵−2411)/16<br>92×17⁵¹³¹¹+1<br>(3963×17⁴⁴⁷³²−11)/16<br>(60381×17³⁷⁰⁹⁶−13)/16||http://factordb.com/index.php?id=1100000008610514108&open=prime<br>http://factordb.com/index.php?id=1100000008610515753&open=prime<br>http://factordb.com/index.php?id=1100000000765961429&open=prime<br>http://factordb.com/index.php?id=1100000000765961411&open=prime<br>http://factordb.com/index.php?id=1100000008610516879&open=prime<br>http://factordb.com/index.php?id=1100000000808118219&open=prime<br>http://factordb.com/index.php?id=1100000003993647842&open=prime<br>http://factordb.com/index.php?id=1100000000765961389&open=prime<br>http://factordb.com/index.php?id=1100000003883765450&open=prime<br>http://factordb.com/index.php?id=1100000003848346668&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008610514108&base=17<br>http://factordb.com/index.php?showid=1100000008610515753&base=17<br>http://factordb.com/index.php?showid=1100000000765961429&base=17<br>http://factordb.com/index.php?showid=1100000000765961411&base=17<br>http://factordb.com/index.php?showid=1100000008610516879&base=17<br>http://factordb.com/index.php?showid=1100000000808118219&base=17<br>http://factordb.com/index.php?showid=1100000003993647842&base=17<br>http://factordb.com/index.php?showid=1100000000765961389&base=17<br>http://factordb.com/index.php?showid=1100000003883765450&base=17<br>http://factordb.com/index.php?showid=1100000003848346668&base=17<nowiki/>||12||200000|| |- ||18||549||C0₆₂₆₈C5<br>H₇₆₆FH<br>80₂₉₈B<br>C0₁₁₆F5<br>HD₉₃<br>GG0₃₀1<br>CF₃₀5<br>B₁₉6B<br>CCF₁₄5<br>7₁₄G7||6271<br>768<br>300<br>119<br>94<br>33<br>32<br>21<br>17<br>16||7872<br>965<br>377<br>150<br>118<br>42<br>41<br>27<br>22<br>20||12×18⁶²⁷⁰+221<br>18⁷⁶⁸−37<br>8×18²⁹⁹+11<br>12×18¹¹⁸+275<br>(302×18⁹³−13)/17<br>304×18³¹+1<br>(219×18³¹−185)/17<br>(11×18²¹−1541)/17<br>(3891×18¹⁵−185)/17<br>(7×18¹⁶+2747)/17||http://factordb.com/index.php?id=1100000003590442437&open=prime<br>http://factordb.com/index.php?id=1100000003590430490&open=prime<br>http://factordb.com/index.php?id=1100000002355574745&open=prime<br>http://factordb.com/index.php?id=1100000002632837015&open=ecm<br>http://factordb.com/index.php?id=1100000002321052894&open=ecm<br>http://factordb.com/index.php?id=1100000000819230161&open=ecm<br>http://factordb.com/index.php?id=1100000002631240657&open=ecm<br>http://factordb.com/index.php?id=1100000003590430474&open=ecm<br>http://factordb.com/index.php?id=1100000003590430470&open=ecm<br>http://factordb.com/index.php?id=1100000003590430465&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003590442437&base=18<br>http://factordb.com/index.php?showid=1100000003590430490&base=18<br>http://factordb.com/index.php?showid=1100000002355574745&base=18<br>http://factordb.com/index.php?showid=1100000002632837015&base=18<br>http://factordb.com/index.php?showid=1100000002321052894&base=18<br>http://factordb.com/index.php?showid=1100000000819230161&base=18<br>http://factordb.com/index.php?showid=1100000002631240657&base=18<br>http://factordb.com/index.php?showid=1100000003590430474&base=18<br>http://factordb.com/index.php?showid=1100000003590430470&base=18<br>http://factordb.com/index.php?showid=1100000003590430465&base=18<nowiki/>||0||–|| |- ||19||31417~31434||1E70₁₂₂₈₉₆1<br>40₁₂₁₈₄₆HB5<br>35₁₂₀₅₆₂<br>FH0H₁₁₂₆₅₉<br>FG6₁₁₀₉₈₄<br>H₈₆₂₉₁6<br>D90₇₃₀₄₆9<br>4F0₄₉₈₄₇6<br>2₄₈₂₂₄7<br>2₄₅₈₈₆7A||122900<br>121850<br>120563<br>112662<br>110986<br>86292<br>73049<br>49850<br>48225<br>45888||157158<br>155816<br>154170<br>144067<br>110347<br>141924<br>93412<br>63746<br>61667<br>58679||634×19¹²²⁸⁹⁷+1<br>4×19¹²¹⁸⁴⁹+6351<br>(59×19¹²⁰⁵⁶²−5)/18<br>(103301×19¹¹²⁶⁵⁹−17)/18<br>(904×19¹¹⁰⁹⁸⁴−1)/3<br>(17×19⁸⁶²⁹²−215)/18<br>256×19⁷³⁰⁴⁷+9<br>91×19⁴⁹⁸⁴⁸+6<br>(19⁴⁸²²⁵+44)/9<br>(19⁴⁵⁸⁸⁸+926)/9||http://factordb.com/index.php?id=1100000001582289581&open=prime<br>http://factordb.com/index.php?id=1100000008755307222&open=prime<br>http://factordb.com/index.php?id=1100000005513825027&open=prime<br>http://factordb.com/index.php?id=1100000008755311453&open=prime<br>http://factordb.com/index.php?id=1100000000808118212&open=prime<br>http://factordb.com/index.php?id=1100000004163040839&open=prime<br>http://factordb.com/index.php?id=1100000003998413751&open=prime<br>http://factordb.com/index.php?id=1100000000808118332&open=prime<br>http://factordb.com/index.php?id=1100000003949188041&open=prime<br>http://factordb.com/index.php?id=1100000003949189035&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000001582289581&base=19<br>http://factordb.com/index.php?showid=1100000008755307222&base=19<br>http://factordb.com/index.php?showid=1100000005513825027&base=19<br>http://factordb.com/index.php?showid=1100000008755311453&base=19<br>http://factordb.com/index.php?showid=1100000000808118212&base=19<br>http://factordb.com/index.php?showid=1100000004163040839&base=19<br>http://factordb.com/index.php?showid=1100000003998413751&base=19<br>http://factordb.com/index.php?showid=1100000000808118332&base=19<br>http://factordb.com/index.php?showid=1100000003949188041&base=19<br>http://factordb.com/index.php?showid=1100000003949189035&base=19<nowiki/>||17||200000|| |- ||20||3314||G0₆₂₆₉D<br>CD₂₄₄₉<br>50₁₁₆₃AJ<br>J₆₅₅05J<br>JCJ₆₂₉<br>E₅₆₆C7<br>3A₅₂₇3<br>G₄₄₇99<br>EC0₄₂₉7<br>40₃₈₇404B||6271<br>2450<br>1166<br>658<br>631<br>568<br>529<br>449<br>432<br>392||8159<br>3188<br>1517<br>857<br>821<br>739<br>688<br>585<br>562<br>510||16×20⁶²⁷⁰+13<br>(241×20²⁴⁴⁹−13)/19<br>5×20¹¹⁶⁵+219<br>20⁶⁵⁸−7881<br>393×20⁶²⁹−1<br>(14×20⁵⁶⁸−907)/19<br>(67×20⁵²⁸−143)/19<br>(16×20⁴⁴⁹−2809)/19<br>292×20⁴³⁰+7<br>4×20³⁹¹+32091||http://factordb.com/index.php?id=1100000003590539457&open=prime<br>http://factordb.com/index.php?id=1100000002325393915&open=prime<br>http://factordb.com/index.php?id=1100000003590502412&open=prime<br>http://factordb.com/index.php?id=1100000003590502490&open=prime<br>http://factordb.com/index.php?id=1100000001559454258&open=prime<br>http://factordb.com/index.php?id=1100000003590502516&open=prime<br>http://factordb.com/index.php?id=1100000003590502531&open=prime<br>http://factordb.com/index.php?id=1100000000840126753&open=prime<br>http://factordb.com/index.php?id=1100000002633348702&open=prime<br>http://factordb.com/index.php?id=1100000003590502563&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003590539457&base=20<br>http://factordb.com/index.php?showid=1100000002325393915&base=20<br>http://factordb.com/index.php?showid=1100000003590502412&base=20<br>http://factordb.com/index.php?showid=1100000003590502490&base=20<br>http://factordb.com/index.php?showid=1100000001559454258&base=20<br>http://factordb.com/index.php?showid=1100000003590502516&base=20<br>http://factordb.com/index.php?showid=1100000003590502531&base=20<br>http://factordb.com/index.php?showid=1100000000840126753&base=20<br>http://factordb.com/index.php?showid=1100000002633348702&base=20<br>http://factordb.com/index.php?showid=1100000003590502563&base=20<nowiki/>||0||–|| |- ||21||13386~13394||27₁₈₄₄₉₉9D<br>F9₁₇₈₇₇₁D<br>2FC₁₁₂₀₂₂A<br>7₁₀₈₄₅₀ID<br>40₄₇₃₃₃9G<br>B90₄₅₀₁₉E5<br>HD₃₇₄₁₄<br>BD₃₅₀₂₇B<br>990₃₃₂₃₉99H<br>5₃₀₆₀₆FEK||184502<br>178773<br>112025<br>108452<br>47336<br>45023<br>37415<br>35029<br>33244<br>30609||243952<br>236377<br>148121<br>143397<br>62588<br>59531<br>49471<br>46316<br>43956<br>40472||(47×21¹⁸⁴⁵⁰¹+953)/20<br>(309×21¹⁷⁸⁷⁷²+71)/20<br>(288×21¹¹²⁰²³−13)/5<br>(7×21¹⁰⁸⁴⁵²+4733)/20<br>4×21⁴⁷³³⁵+205<br>240×21⁴⁵⁰²¹+299<br>(353×21³⁷⁴¹⁴−13)/20<br>(233×21³⁵⁰²⁸−53)/20<br>198×21³³²⁴²+4175<br>(21³⁰⁶⁰⁹+18455)/4||http://factordb.com/index.php?id=1100000008700600990&open=prime<br>http://factordb.com/index.php?id=1100000008700596669&open=prime<br>http://factordb.com/index.php?id=1100000008700593358&open=prime<br>http://factordb.com/index.php?id=1100000008700586183&open=prime<br>http://factordb.com/index.php?id=1100000000808118331&open=prime<br>http://factordb.com/index.php?id=1100000003996110311&open=prime<br>http://factordb.com/index.php?id=1100000003996110479&open=prime<br>http://factordb.com/index.php?id=1100000003996110718&open=prime<br>http://factordb.com/index.php?id=1100000003996110944&open=prime<br>http://factordb.com/index.php?id=1100000003996111130&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008700600990&base=21<br>http://factordb.com/index.php?showid=1100000008700596669&base=21<br>http://factordb.com/index.php?showid=1100000008700593358&base=21<br>http://factordb.com/index.php?showid=1100000008700586183&base=21<br>http://factordb.com/index.php?showid=1100000000808118331&base=21<br>http://factordb.com/index.php?showid=1100000003996110311&base=21<br>http://factordb.com/index.php?showid=1100000003996110479&base=21<br>http://factordb.com/index.php?showid=1100000003996110718&base=21<br>http://factordb.com/index.php?showid=1100000003996110944&base=21<br>http://factordb.com/index.php?showid=1100000003996111130&base=21<nowiki/>||8||200000|| |- ||22||8003||BK₂₂₀₀₁5<br>7₃₈₁₅2L<br>L₂₃₈₅KE7<br>7₉₅₉K7<br>J0₇₆₇IGGJ<br>K0₇₆₀EC1<br>I₆₂₆AF<br>E60₄₉₆L<br>L₄₈₃G3<br>L0₄₅₄B63||22003<br>3817<br>2388<br>961<br>772<br>764<br>628<br>499<br>485<br>458||29538<br>5124<br>3206<br>1290<br>1037<br>1026<br>843<br>670<br>652<br>615||(251×22²²⁰⁰²−335)/21<br>(22³⁸¹⁷−289)/3<br>22²³⁸⁸−653<br>(22⁹⁶¹+857)/3<br>19×22⁷⁷¹+199779<br>20×22⁷⁶³+7041<br>(6×22⁶²⁸−1259)/7<br>314×22⁴⁹⁷+21<br>22⁴⁸⁵−129<br>21×22⁴⁵⁷+5459||http://factordb.com/index.php?id=1100000003594696838&open=prime<br>http://factordb.com/index.php?id=1100000003591359839&open=prime<br>http://factordb.com/index.php?id=1100000003591360774&open=prime<br>http://factordb.com/index.php?id=1100000003591361817&open=prime<br>http://factordb.com/index.php?id=1100000003591362567&open=prime<br>http://factordb.com/index.php?id=1100000000632724415&open=prime<br>http://factordb.com/index.php?id=1100000000632724334&open=prime<br>http://factordb.com/index.php?id=1100000000632703239&open=prime<br>http://factordb.com/index.php?id=1100000003591364730&open=prime<br>http://factordb.com/index.php?id=1100000003591365331&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003594696838&base=22<br>http://factordb.com/index.php?showid=1100000003591359839&base=22<br>http://factordb.com/index.php?showid=1100000003591360774&base=22<br>http://factordb.com/index.php?showid=1100000003591361817&base=22<br>http://factordb.com/index.php?showid=1100000003591362567&base=22<br>http://factordb.com/index.php?showid=1100000000632724415&base=22<br>http://factordb.com/index.php?showid=1100000000632724334&base=22<br>http://factordb.com/index.php?showid=1100000000632703239&base=22<br>http://factordb.com/index.php?showid=1100000003591364730&base=22<br>http://factordb.com/index.php?showid=1100000003591365331&base=22<nowiki/>||0||–|| |- ||23||65178~65265||B0₉₃₀₄₆FB<br>L₈₆₄₄₄D<br>AJ₈₁₀₆₅4<br>20₇₃₅₆₀98<br>J₆₈₂₁₇G4<br>D70₆₆₇₇₀B<br>5F₆₂₃₄₀6<br>A7M7₆₁₅₃₂<br>B30₆₁₁₃₆5<br>EJ₅₂₁₆₉||93049<br>86445<br>81067<br>73563<br>68219<br>66773<br>62342<br>61535<br>61139<br>52170||126708<br>117715<br>110391<br>100172<br>92896<br>90927<br>84893<br>83794<br>83255<br>71042||11×23⁹³⁰⁴⁸+356<br>(21×23⁸⁶⁴⁴⁵−197)/22<br>(239×23⁸¹⁰⁶⁶−349)/22<br>2×23⁷³⁵⁶²+215<br>(19×23⁶⁸²¹⁹−1867)/22<br>306×23⁶⁶⁷⁷¹+11<br>(125×23⁶²³⁴¹−213)/22<br>(120413×23⁶¹⁵³²−7)/22<br>256×23⁶¹¹³⁷+5<br>(327×23⁵²¹⁶⁹−19)/22||http://factordb.com/index.php?id=1100000004691540361&open=prime<br>http://factordb.com/index.php?id=1100000004691546739&open=prime<br>http://factordb.com/index.php?id=1100000004691548070&open=prime<br>http://factordb.com/index.php?id=1100000004691548569&open=prime<br>http://factordb.com/index.php?id=1100000004691549462&open=prime<br>http://factordb.com/index.php?id=1100000004691549803&open=prime<br>http://factordb.com/index.php?id=1100000004691551005&open=prime<br>http://factordb.com/index.php?id=1100000004691556967&open=prime<br>http://factordb.com/index.php?id=1100000004691557254&open=prime<br>http://factordb.com/index.php?id=1100000004691557548&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004691540361&base=23<br>http://factordb.com/index.php?showid=1100000004691546739&base=23<br>http://factordb.com/index.php?showid=1100000004691548070&base=23<br>http://factordb.com/index.php?showid=1100000004691548569&base=23<br>http://factordb.com/index.php?showid=1100000004691549462&base=23<br>http://factordb.com/index.php?showid=1100000004691549803&base=23<br>http://factordb.com/index.php?showid=1100000004691551005&base=23<br>http://factordb.com/index.php?showid=1100000004691556967&base=23<br>http://factordb.com/index.php?showid=1100000004691557254&base=23<br>http://factordb.com/index.php?showid=1100000004691557548&base=23<nowiki/>||87||100000|| |- ||24||3409||N00N₈₁₂₉LN<br>88N₅₉₅₁<br>A0₂₉₅₁8ID<br>D₂₆₉₈LD<br>N₂₆₄₄LLN<br>BC0₃₃₁B<br>20₃₁₃7<br>C7₂₉₈<br>D0₂₅₉KKD<br>I0₂₄₁I5||8134<br>5953<br>2955<br>2700<br>2647<br>334<br>315<br>299<br>263<br>244||11227<br>8216<br>4079<br>3727<br>3654<br>461<br>434<br>413<br>363<br>337||13249×24⁸¹³¹−49<br>201×24⁵⁹⁵¹−1<br>10×24²⁹⁵⁴+5053<br>(13×24²⁷⁰⁰+4403)/23<br>24²⁶⁴⁷−1201<br>276×24³³²+11<br>2×24³¹⁴+7<br>(283×24²⁹⁸−7)/23<br>13×24²⁶²+12013<br>18×24²⁴³+437||http://factordb.com/index.php?id=1100000003593391606&open=prime<br>http://factordb.com/index.php?id=1100000003593275880&open=prime<br>http://factordb.com/index.php?id=1100000003593269654&open=prime<br>http://factordb.com/index.php?id=1100000003593269876&open=prime<br>http://factordb.com/index.php?id=1100000003593270089&open=prime<br>http://factordb.com/index.php?id=1100000002633359842&open=prime<br>http://factordb.com/index.php?id=1100000002355610241&open=prime<br>http://factordb.com/index.php?id=1100000002326181235&open=prime<br>http://factordb.com/index.php?id=1100000003593270725&open=prime<br>http://factordb.com/index.php?id=1100000002633360037&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003593391606&base=24<br>http://factordb.com/index.php?showid=1100000003593275880&base=24<br>http://factordb.com/index.php?showid=1100000003593269654&base=24<br>http://factordb.com/index.php?showid=1100000003593269876&base=24<br>http://factordb.com/index.php?showid=1100000003593270089&base=24<br>http://factordb.com/index.php?showid=1100000002633359842&base=24<br>http://factordb.com/index.php?showid=1100000002355610241&base=24<br>http://factordb.com/index.php?showid=1100000002326181235&base=24<br>http://factordb.com/index.php?showid=1100000003593270725&base=24<br>http://factordb.com/index.php?showid=1100000002633360037&base=24<nowiki/>||0||–|| |- ||25||133639~133724||E₉₈₃₉₆FOO<br>1J710₉₆₂₇₂1<br>NB0₈₅₅₉₈5NH<br>D70₈₁₅₈₁JJ7<br>F0₈₀₀₅₄HL<br>J010₇₅₉₄₃E7<br>K₆₇₇₇₁5I<br>LO₆₆₃₇₇KC<br>KJD0₆₃₃₉₉1<br>70₆₀₈₉₂D711||98399<br>96277<br>85603<br>81586<br>80057<br>75948<br>67773<br>66380<br>63403<br>60897||137556<br>134589<br>119668<br>114053<br>111915<br>106171<br>94743<br>92796<br>88634<br>85130||(7×25⁹⁸³⁹⁹+10613)/12<br>27676×25⁹⁶²⁷³+1<br>586×25⁸⁵⁶⁰¹+3717<br>332×25⁸¹⁵⁸⁴+12357<br>15×25⁸⁰⁰⁵⁶+446<br>11876×25⁷⁵⁹⁴⁵+357<br>(5×25⁶⁷⁷⁷³−2267)/6<br>22×25⁶⁶³⁷⁹−113<br>12988×25⁶³⁴⁰⁰+1<br>7×25⁶⁰⁸⁹⁶+207526||http://factordb.com/index.php?id=1100000000808118215&open=prime<br>http://factordb.com/index.php?id=1100000003983674902&open=prime<br>http://factordb.com/index.php?id=1100000004909706420&open=prime<br>http://factordb.com/index.php?id=1100000004909733266&open=prime<br>http://factordb.com/index.php?id=1100000004909750102&open=prime<br>http://factordb.com/index.php?id=1100000004909770736&open=prime<br>http://factordb.com/index.php?id=1100000004586986394&open=prime<br>http://factordb.com/index.php?id=1100000000808118270&open=prime<br>http://factordb.com/index.php?id=1100000004586986664&open=prime<br>http://factordb.com/index.php?id=1100000004586986798&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118215&base=25<br>http://factordb.com/index.php?showid=1100000003983674902&base=25<br>http://factordb.com/index.php?showid=1100000004909706420&base=25<br>http://factordb.com/index.php?showid=1100000004909733266&base=25<br>http://factordb.com/index.php?showid=1100000004909750102&base=25<br>http://factordb.com/index.php?showid=1100000004909770736&base=25<br>http://factordb.com/index.php?showid=1100000004586986394&base=25<br>http://factordb.com/index.php?showid=1100000000808118270&base=25<br>http://factordb.com/index.php?showid=1100000004586986664&base=25<br>http://factordb.com/index.php?showid=1100000004586986798&base=25<nowiki/>||85||100000|| |- ||26||25256~25259||85M₁₉₇₀₆₀B<br>M0₆₁₁₈₆2BB<br>J0₄₄₃₀₃KCB<br>6K₂₃₃₀₀5<br>LD0₂₀₉₇₅7<br>7₂₀₂₇₉OL<br>5₁₉₃₉₁6F<br>9GDK₁₅₉₂₀P<br>M₈₇₇₂P<br>K0₄₃₆₄I5||197063<br>61190<br>44307<br>23302<br>20978<br>20281<br>19393<br>15924<br>8773<br>4367||278839<br>86583<br>62694<br>32972<br>29684<br>28697<br>27440<br>22532<br>12414<br>6180||(5347×26¹⁹⁷⁰⁶¹−297)/25<br>22×26⁶¹¹⁸⁹+1649<br>19×26⁴⁴³⁰⁶+13843<br>(34×26²³³⁰¹−79)/5<br>559×26²⁰⁹⁷⁶+7<br>(7×26²⁰²⁸¹+11393)/25<br>(26¹⁹³⁹³+179)/5<br>(32569×26¹⁵⁹²¹+21)/5<br>(22×26⁸⁷⁷³+53)/25<br>20×26⁴³⁶⁶+473||http://factordb.com/index.php?id=1100000008573990023&open=prime<br>http://factordb.com/index.php?id=1100000003968169875&open=prime<br>http://factordb.com/index.php?id=1100000003968156595&open=prime<br>http://factordb.com/index.php?id=1100000003892628745&open=prime<br>http://factordb.com/index.php?id=1100000003892628658&open=prime<br>http://factordb.com/index.php?id=1100000003892628605&open=prime<br>http://factordb.com/index.php?id=1100000003850151202&open=prime<br>http://factordb.com/index.php?id=1100000003850155316&open=prime<br>http://factordb.com/index.php?id=1100000000758011195&open=prime<br>http://factordb.com/index.php?id=1100000002634136508&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008573990023&base=26<br>http://factordb.com/index.php?showid=1100000003968169875&base=26<br>http://factordb.com/index.php?showid=1100000003968156595&base=26<br>http://factordb.com/index.php?showid=1100000003892628745&base=26<br>http://factordb.com/index.php?showid=1100000003892628658&base=26<br>http://factordb.com/index.php?showid=1100000003892628605&base=26<br>http://factordb.com/index.php?showid=1100000003850151202&base=26<br>http://factordb.com/index.php?showid=1100000003850155316&base=26<br>http://factordb.com/index.php?showid=1100000000758011195&base=26<br>http://factordb.com/index.php?showid=1100000002634136508&base=26<nowiki/>||3||200000|| |- ||27||102852~102896||CA0F₈₈₈₈₃A<br>GNN0₇₈₇₉₅N<br>O44L₆₆₀₁₆7<br>NJ0₆₄₃₆₉H<br>ME₄₉₆₄₀9G<br>PH0₄₇₈₉₀1<br>QF₄₇₁₆₅AF5<br>J0₄₀₇₉₁PD<br>510₃₉₁₆₄I07<br>NGN0₃₆₃₂₉N||88887<br>78799<br>66020<br>64372<br>49643<br>47893<br>47169<br>40794<br>39169<br>36333||127230<br>112790<br>94499<br>92140<br>71058<br>68553<br>67516<br>58391<br>56065<br>52006||(234483×27⁸⁸⁸⁸⁴−145)/26<br>12308×27⁷⁸⁷⁹⁶+23<br>(457829×27⁶⁶⁰¹⁷−385)/26<br>640×27⁶⁴³⁷⁰+17<br>(293×27⁴⁹⁶⁴²−1736)/13<br>692×27⁴⁷⁸⁹¹+1<br>(691×27⁴⁷¹⁶⁸−95045)/26<br>19×27⁴⁰⁷⁹³+688<br>136×27³⁹¹⁶⁷+13129<br>17222×27³⁶³³⁰+23||http://factordb.com/index.php?id=1100000000808118233&open=prime<br>http://factordb.com/index.php?id=1100000004681348398&open=prime<br>http://factordb.com/index.php?id=1100000004374140861&open=prime<br>http://factordb.com/index.php?id=1100000004374138999&open=prime<br>http://factordb.com/index.php?id=1100000000819229859&open=prime<br>http://factordb.com/index.php?id=1100000004102754118&open=prime<br>http://factordb.com/index.php?id=1100000004102755880&open=prime<br>http://factordb.com/index.php?id=1100000004102758254&open=prime<br>http://factordb.com/index.php?id=1100000004102875088&open=prime<br>http://factordb.com/index.php?id=1100000004103372866&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118233&base=27<br>http://factordb.com/index.php?showid=1100000004681348398&base=27<br>http://factordb.com/index.php?showid=1100000004374140861&base=27<br>http://factordb.com/index.php?showid=1100000004374138999&base=27<br>http://factordb.com/index.php?showid=1100000000819229859&base=27<br>http://factordb.com/index.php?showid=1100000004102754118&base=27<br>http://factordb.com/index.php?showid=1100000004102755880&base=27<br>http://factordb.com/index.php?showid=1100000004102758254&base=27<br>http://factordb.com/index.php?showid=1100000004102875088&base=27<br>http://factordb.com/index.php?showid=1100000004103372866&base=27<nowiki/>||44||100000|| |- ||28||25528~25529||O4O₉₄₅₃₅9<br>5OA₃₁₂₃₈F<br>N6₂₄₀₅₁LR<br>D0₅₂₆₇77D<br>QO₄₂₃₉69<br>5₃₇₄₆8P<br>G0₁₈₉₉AN<br>A₁₄₂₃6F<br>5I₁₃₇₀F<br>5₁₃₃₂P8P||94538<br>31241<br>24054<br>5271<br>4242<br>3748<br>1902<br>1425<br>1372<br>1335||136812<br>45210<br>34810<br>7628<br>6139<br>5424<br>2753<br>2062<br>1985<br>1932||(6092×28⁹⁴⁵³⁶−143)/9<br>(4438×28³¹²³⁹+125)/27<br>(209×28²⁴⁰⁵³+3967)/9<br>13×28⁵²⁷⁰+5697<br>(242×28⁴²⁴¹−4679)/9<br>(5×28³⁷⁴⁸+2803)/27<br>16×28¹⁹⁰¹+303<br>(10×28¹⁴²⁵−2899)/27<br>(17×28¹³⁷¹−11)/3<br>(5×28¹³³⁵+426163)/27||http://factordb.com/index.php?id=1100000000808118231&open=prime<br>http://factordb.com/index.php?id=1100000003880455200&open=prime<br>http://factordb.com/index.php?id=1100000003879667576&open=prime<br>http://factordb.com/index.php?id=1100000003850151420&open=prime<br>http://factordb.com/index.php?id=1100000000840839934&open=prime<br>http://factordb.com/index.php?id=1100000003850161974&open=prime<br>http://factordb.com/index.php?id=1100000003850161973&open=prime<br>http://factordb.com/index.php?id=1100000000840839947&open=prime<br>http://factordb.com/index.php?id=1100000003850161972&open=prime<br>http://factordb.com/index.php?id=1100000003850161965&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118231&base=28<br>http://factordb.com/index.php?showid=1100000003880455200&base=28<br>http://factordb.com/index.php?showid=1100000003879667576&base=28<br>http://factordb.com/index.php?showid=1100000003850151420&base=28<br>http://factordb.com/index.php?showid=1100000000840839934&base=28<br>http://factordb.com/index.php?showid=1100000003850161974&base=28<br>http://factordb.com/index.php?showid=1100000003850161973&base=28<br>http://factordb.com/index.php?showid=1100000000840839947&base=28<br>http://factordb.com/index.php?showid=1100000003850161972&base=28<br>http://factordb.com/index.php?showid=1100000003850161965&base=28<nowiki/>||1||900000|| |- ||29||355242~355367||830₉₉₃₇₇4<br>GP5J₉₄₉₃₅<br>P05J₉₀₂₈₉<br>BBD0₈₈₈₈₈PB<br>8B₈₅₃₃₃G<br>L0₈₁₅₇₁5955<br>E0₇₇₃₇₂L7B<br>LPC₇₅₁₅₁9<br>JR0₇₄₆₂₂7<br>B₇₄₅₀₁0RP||99380<br>94938<br>90292<br>88893<br>85335<br>81576<br>77376<br>75154<br>74625<br>74504||145333<br>138837<br>132043<br>129997<br>124794<br>119297<br>113155<br>109905<br>109132<br>108955||235×29⁹⁹³⁷⁸+4<br>(397227×29⁹⁴⁹³⁵−19)/28<br>(588859×29⁹⁰²⁸⁹−19)/28<br>9583×29⁸⁸⁸⁹⁰+736<br>(235×29⁸⁵³³⁴+129)/14<br>21×29⁸¹⁵⁷⁵+129664<br>14×29⁷⁷³⁷⁵+17875<br>(4441×29⁷⁵¹⁵²−24)/7<br>578×29⁷⁴⁶²³+7<br>(11×29⁷⁴⁵⁰⁴−245655)/28||http://factordb.com/index.php?id=1100000008253882372&open=prime<br>http://factordb.com/index.php?id=1100000008253893542&open=prime<br>http://factordb.com/index.php?id=1100000008253899083&open=prime<br>http://factordb.com/index.php?id=1100000008253909183&open=prime<br>http://factordb.com/index.php?id=1100000008253921388&open=prime<br>http://factordb.com/index.php?id=1100000008253925955&open=prime<br>http://factordb.com/index.php?id=1100000008253931446&open=prime<br>http://factordb.com/index.php?id=1100000000808118236&open=prime<br>http://factordb.com/index.php?id=1100000008253934219&open=prime<br>http://factordb.com/index.php?id=1100000008253936120&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008253882372&base=29<br>http://factordb.com/index.php?showid=1100000008253893542&base=29<br>http://factordb.com/index.php?showid=1100000008253899083&base=29<br>http://factordb.com/index.php?showid=1100000008253909183&base=29<br>http://factordb.com/index.php?showid=1100000008253921388&base=29<br>http://factordb.com/index.php?showid=1100000008253925955&base=29<br>http://factordb.com/index.php?showid=1100000008253931446&base=29<br>http://factordb.com/index.php?showid=1100000000808118236&base=29<br>http://factordb.com/index.php?showid=1100000008253934219&base=29<br>http://factordb.com/index.php?showid=1100000008253936120&base=29<nowiki/>||125||100000|| |- ||30||2619||OT₃₄₂₀₅<br>I0₂₄₆₀₈D<br>5₄₈₈₂J<br>C0₁₀₂₂1<br>M0₅₄₇SS7<br>M₂₄₁QB<br>AN₂₀₆<br>50₁₆₄B<br>J₁₅₃QJ<br>J₉₄QQJ||34206<br>24610<br>4883<br>1024<br>551<br>243<br>207<br>166<br>155<br>97||50527<br>36352<br>7213<br>1513<br>814<br>359<br>306<br>245<br>229<br>144||25×30³⁴²⁰⁵−1<br>18×30²⁴⁶⁰⁹+13<br>(5×30⁴⁸⁸³+401)/29<br>12×30¹⁰²³+1<br>22×30⁵⁵⁰+26047<br>(22×30²⁴³+3139)/29<br>(313×30²⁰⁶−23)/29<br>5×30¹⁶⁵+11<br>(19×30¹⁵⁵+6071)/29<br>(19×30⁹⁷+188771)/29||http://factordb.com/index.php?id=1100000000800812865&open=prime<br>http://factordb.com/index.php?id=1100000003593967511&open=prime<br>http://factordb.com/index.php?id=1100000002327649423&open=prime<br>http://factordb.com/index.php?id=1100000000785448736&open=prime<br>http://factordb.com/index.php?id=1100000003593407988&open=prime<br>http://factordb.com/index.php?id=1100000003593408295&open=prime<br>http://factordb.com/index.php?id=1100000002327651073&open=prime<br>http://factordb.com/index.php?id=1100000002356282476&open=ecm<br>http://factordb.com/index.php?id=1100000003593409109&open=ecm<br>http://factordb.com/index.php?id=1100000003593409165&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000800812865&base=30<br>http://factordb.com/index.php?showid=1100000003593967511&base=30<br>http://factordb.com/index.php?showid=1100000002327649423&base=30<br>http://factordb.com/index.php?showid=1100000000785448736&base=30<br>http://factordb.com/index.php?showid=1100000003593407988&base=30<br>http://factordb.com/index.php?showid=1100000003593408295&base=30<br>http://factordb.com/index.php?showid=1100000002327651073&base=30<br>http://factordb.com/index.php?showid=1100000002356282476&base=30<br>http://factordb.com/index.php?showid=1100000003593409109&base=30<br>http://factordb.com/index.php?showid=1100000003593409165&base=30<nowiki/>||0||–|| |- ||31||569323~569400||2IIF₉₁₈₀₅<br>B0₈₈₃₀₉APO9<br>J0T₇₇₅₁₆<br>J090₇₇₁₂₈NNN<br>D₆₉₈₆₁QO<br>9MH0₆₈₆₃₇D<br>J₆₇₁₆₂D<br>N0₆₆₉₇₁32P<br>DDDQ0₆₄₀₈₈TD<br>U₆₃₈₆₁CM3||91808<br>88314<br>77518<br>77134<br>69863<br>68641<br>67163<br>66975<br>64094<br>63864||136918<br>131708<br>115608<br>115035<br>104191<br>102369<br>100165<br>99884<br>95587<br>95245||(4997×31⁹¹⁸⁰⁵−1)/2<br>11×31⁸⁸³¹³+322688<br>(17699×31⁷⁷⁵¹⁶−29)/30<br>18268×31⁷⁷¹³¹+22839<br>(13×31⁶⁹⁸⁶³+12407)/30<br>9348×31⁶⁸⁶³⁸+13<br>(19×31⁶⁷¹⁶³−199)/30<br>23×31⁶⁶⁹⁷⁴+2970<br>400205×31⁶⁴⁰⁹⁰+912<br>31⁶³⁸⁶⁴−17574||http://factordb.com/index.php?id=1100000007050395732&open=prime<br>http://factordb.com/index.php?id=1100000007050397309&open=prime<br>http://factordb.com/index.php?id=1100000007050398940&open=prime<br>http://factordb.com/index.php?id=1100000007050400178&open=prime<br>http://factordb.com/index.php?id=1100000006965878559&open=prime<br>http://factordb.com/index.php?id=1100000006965875678&open=prime<br>http://factordb.com/index.php?id=1100000006965873668&open=prime<br>http://factordb.com/index.php?id=1100000006965870538&open=prime<br>http://factordb.com/index.php?id=1100000006965868103&open=prime<br>http://factordb.com/index.php?id=1100000006965865343&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000007050395732&base=31<br>http://factordb.com/index.php?showid=1100000007050397309&base=31<br>http://factordb.com/index.php?showid=1100000007050398940&base=31<br>http://factordb.com/index.php?showid=1100000007050400178&base=31<br>http://factordb.com/index.php?showid=1100000006965878559&base=31<br>http://factordb.com/index.php?showid=1100000006965875678&base=31<br>http://factordb.com/index.php?showid=1100000006965873668&base=31<br>http://factordb.com/index.php?showid=1100000006965870538&base=31<br>http://factordb.com/index.php?showid=1100000006965868103&base=31<br>http://factordb.com/index.php?showid=1100000006965865343&base=31<nowiki/>||77||100000|| |- ||32||168882~169002||V₉₉₅₈₃63<br>6₈₉₀₇₄AF<br>8₇₇₇₀₀H<br>Q₇₇₄₀₁EQQQ3<br>8₇₇₂₄₉3<br>JM₇₆₀₂₈L<br>E₇₂₉₁₉IL<br>B0₆₇₆₈₀CB<br>GK₆₆₀₇₆F<br>KN₆₅₀₂₂||99585<br>89076<br>77701<br>77406<br>77250<br>76030<br>72921<br>67683<br>66078<br>65023||149891<br>134073<br>116952<br>116508<br>116273<br>114437<br>109757<br>101873<br>99458<br>97870||32⁹⁹⁵⁸⁵−829<br>(6×32⁸⁹⁰⁷⁶+4241)/31<br>(8×32⁷⁷⁷⁰¹+271)/31<br>(26×32⁷⁷⁴⁰⁶−390071011)/31<br>(8×32⁷⁷²⁵⁰−163)/31<br>(611×32⁷⁶⁰²⁹−53)/31<br>(14×32⁷²⁹²¹+4171)/31<br>11×32⁶⁷⁶⁸²+395<br>(516×32⁶⁶⁰⁷⁷−175)/31<br>(643×32⁶⁵⁰²²−23)/31||http://factordb.com/index.php?id=1100000005514892191&open=prime<br>http://factordb.com/index.php?id=1100000005514897129&open=prime<br>http://factordb.com/index.php?id=1100000005514901700&open=prime<br>http://factordb.com/index.php?id=1100000005514915338&open=prime<br>http://factordb.com/index.php?id=1100000005514918574&open=prime<br>http://factordb.com/index.php?id=1100000005514922523&open=prime<br>http://factordb.com/index.php?id=1100000004591654373&open=prime<br>http://factordb.com/index.php?id=1100000004591654467&open=prime<br>http://factordb.com/index.php?id=1100000004591654632&open=prime<br>http://factordb.com/index.php?id=1100000004591654952&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005514892191&base=32<br>http://factordb.com/index.php?showid=1100000005514897129&base=32<br>http://factordb.com/index.php?showid=1100000005514901700&base=32<br>http://factordb.com/index.php?showid=1100000005514915338&base=32<br>http://factordb.com/index.php?showid=1100000005514918574&base=32<br>http://factordb.com/index.php?showid=1100000005514922523&base=32<br>http://factordb.com/index.php?showid=1100000004591654373&base=32<br>http://factordb.com/index.php?showid=1100000004591654467&base=32<br>http://factordb.com/index.php?showid=1100000004591654632&base=32<br>http://factordb.com/index.php?showid=1100000004591654952&base=32<nowiki/>||120||100000|| |- ||33||280012~280093||DP₉₅₀₉₃M5<br>HJ0₉₄₂₉₅J<br>90₉₃₅₉₇Q<br>9F0₉₃₁₅₇N<br>7₈₉₄₄₉333H<br>K3₈₀₇₅₁6K<br>D₈₀₁₀₇9UD<br>VFU₇₂₂₀₄FK<br>J₆₈₇₁₅2BJ<br>DF0₆₈₃₆₇J||95096<br>94298<br>93599<br>93160<br>89453<br>80754<br>80110<br>72208<br>68718<br>68370||144405<br>143193<br>142131<br>141465<br>135835<br>122626<br>121648<br>109649<br>104350<br>103821||(441×33⁹⁵⁰⁹⁵−3833)/32<br>580×33⁹⁴²⁹⁶+19<br>9×33⁹³⁵⁹⁸+26<br>312×33⁹³¹⁵⁸+23<br>(7×33⁸⁹⁴⁵³−4743239)/32<br>(643×33⁸⁰⁷⁵³+3709)/32<br>(13×33⁸⁰¹¹⁰−121453)/32<br>(16623×33⁷²²⁰⁶−8095)/16<br>(19×33⁶⁸⁷¹⁸−600883)/32<br>444×33⁶⁸³⁶⁸+19||http://factordb.com/index.php?id=1100000005652348775&open=prime<br>http://factordb.com/index.php?id=1100000005652362811&open=prime<br>http://factordb.com/index.php?id=1100000005652375073&open=prime<br>http://factordb.com/index.php?id=1100000005652389776&open=prime<br>http://factordb.com/index.php?id=1100000005652430746&open=prime<br>http://factordb.com/index.php?id=1100000005652446200&open=prime<br>http://factordb.com/index.php?id=1100000005652461592&open=prime<br>http://factordb.com/index.php?id=1100000004614764298&open=prime<br>http://factordb.com/index.php?id=1100000004614770536&open=prime<br>http://factordb.com/index.php?id=1100000004614784274&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005652348775&base=33<br>http://factordb.com/index.php?showid=1100000005652362811&base=33<br>http://factordb.com/index.php?showid=1100000005652375073&base=33<br>http://factordb.com/index.php?showid=1100000005652389776&base=33<br>http://factordb.com/index.php?showid=1100000005652430746&base=33<br>http://factordb.com/index.php?showid=1100000005652446200&base=33<br>http://factordb.com/index.php?showid=1100000005652461592&base=33<br>http://factordb.com/index.php?showid=1100000004614764298&base=33<br>http://factordb.com/index.php?showid=1100000004614770536&base=33<br>http://factordb.com/index.php?showid=1100000004614784274&base=33<nowiki/>||81||100000|| |- ||34||184785~184832||GFGC₉₉₉₉₆5<br>90₉₇₉₅₀FJ<br>NM0₈₅₂₁₈KX<br>F₈₃₁₈₉H2HP<br>P₇₉₄₄₁444P<br>6₇₇₀₂₇8X<br>XQIQ₇₂₂₄₁D<br>T₆₆₅₃₀IF<br>4₆₆₁₅₂B<br>2EEC₆₆₀₃₉7||100000<br>97953<br>85222<br>83193<br>79445<br>77029<br>72245<br>66532<br>66153<br>66043||153148<br>150013<br>130516<br>127408<br>121669<br>117968<br>110642<br>101893<br>101312<br>101143||(209246×34⁹⁹⁹⁹⁷−81)/11<br>9×34⁹⁷⁹⁵²+529<br>804×34⁸⁵²²⁰+713<br>(5×34⁸³¹⁹³+700233)/11<br>(25×34⁷⁹⁴⁴⁵−28062367)/33<br>(2×34⁷⁷⁰²⁹+1043)/11<br>(1288676×34⁷²²⁴²−455)/33<br>(29×34⁶⁶⁵³²−12833)/33<br>(4×34⁶⁶¹⁵³+227)/33<br>(30826×34⁶⁶⁰⁴⁰−59)/11||http://factordb.com/index.php?id=1100000004702891268&open=prime<br>http://factordb.com/index.php?id=1100000004702894713&open=prime<br>http://factordb.com/index.php?id=1100000004702900996&open=prime<br>http://factordb.com/index.php?id=1100000004702910376&open=prime<br>http://factordb.com/index.php?id=1100000004702913746&open=prime<br>http://factordb.com/index.php?id=1100000004702918600&open=prime<br>http://factordb.com/index.php?id=1100000004399656529&open=prime<br>http://factordb.com/index.php?id=1100000004399657696&open=prime<br>http://factordb.com/index.php?id=1100000004399658651&open=prime<br>http://factordb.com/index.php?id=1100000004399659716&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004702891268&base=34<br>http://factordb.com/index.php?showid=1100000004702894713&base=34<br>http://factordb.com/index.php?showid=1100000004702900996&base=34<br>http://factordb.com/index.php?showid=1100000004702910376&base=34<br>http://factordb.com/index.php?showid=1100000004702913746&base=34<br>http://factordb.com/index.php?showid=1100000004702918600&base=34<br>http://factordb.com/index.php?showid=1100000004399656529&base=34<br>http://factordb.com/index.php?showid=1100000004399657696&base=34<br>http://factordb.com/index.php?showid=1100000004399658651&base=34<br>http://factordb.com/index.php?showid=1100000004399659716&base=34<nowiki/>||47||100000|| |- ||35||720002~720062||N0N₉₉₉₇₁9<br>V0₈₃₆₆₉E73<br>N₈₁₅₆₃K7N<br>BJ0₈₁₂₇₉N<br>J0₈₀₀₆₂FUH<br>43V₇₉₇₅₄<br>9₇₆₆₀₀K3<br>LB₇₁₃₆₆PB<br>Q₆₄₁₅₀H<br>50₆₃₃₉₇5R||99974<br>83673<br>81566<br>81282<br>80066<br>79756<br>76602<br>71369<br>64151<br>63400||154367<br>129197<br>125944<br>125505<br>123628<br>123148<br>118279<br>110199<br>99054<br>97894||(27393×35⁹⁹⁹⁷²−499)/34<br>31×35⁸³⁶⁷²+17398<br>(23×35⁸¹⁵⁶⁶−144013)/34<br>404×35⁸¹²⁸⁰+23<br>19×35⁸⁰⁰⁶⁵+19442<br>(4893×35⁷⁹⁷⁵⁴−31)/34<br>(9×35⁷⁶⁶⁰²+12877)/34<br>(725×35⁷¹³⁶⁸+16649)/34<br>(13×35⁶⁴¹⁵¹−166)/17<br>5×35⁶³³⁹⁹+202||http://factordb.com/index.php?id=1100000008248342445&open=prime<br>http://factordb.com/index.php?id=1100000008248353306&open=prime<br>http://factordb.com/index.php?id=1100000008248375642&open=prime<br>http://factordb.com/index.php?id=1100000008248397018&open=prime<br>http://factordb.com/index.php?id=1100000008248412468&open=prime<br>http://factordb.com/index.php?id=1100000008248418540&open=prime<br>http://factordb.com/index.php?id=1100000008248423670&open=prime<br>http://factordb.com/index.php?id=1100000008192119974&open=prime<br>http://factordb.com/index.php?id=1100000008192126630&open=prime<br>http://factordb.com/index.php?id=1100000008192129294&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008248342445&base=35<br>http://factordb.com/index.php?showid=1100000008248353306&base=35<br>http://factordb.com/index.php?showid=1100000008248375642&base=35<br>http://factordb.com/index.php?showid=1100000008248397018&base=35<br>http://factordb.com/index.php?showid=1100000008248412468&base=35<br>http://factordb.com/index.php?showid=1100000008248418540&base=35<br>http://factordb.com/index.php?showid=1100000008248423670&base=35<br>http://factordb.com/index.php?showid=1100000008192119974&base=35<br>http://factordb.com/index.php?showid=1100000008192126630&base=35<br>http://factordb.com/index.php?showid=1100000008192129294&base=35<nowiki/>||60||100000|| |- ||36||35286~35290||P₈₁₉₉₃SZ<br>S0₇₅₀₀₇8H<br>7K₂₆₅₆₇Z<br>J₁₀₁₁₇LJ<br>VL0₇₂₅₈J<br>EO0₆₁₇₇V<br>FZ₅₇₇₇3P<br>T09₄₆₁₈1<br>RY₄₅₆₂H<br>OZ₃₉₃₂AZ||81995<br>75010<br>26569<br>10119<br>7261<br>6180<br>5780<br>4621<br>4564<br>3935||127609<br>116739<br>41349<br>15748<br>11301<br>9618<br>8996<br>7192<br>7103<br>6124||(5×36⁸¹⁹⁹⁵+821)/7<br>28×36⁷⁵⁰⁰⁹+305<br>(53×36²⁶⁵⁶⁸+101)/7<br>(19×36¹⁰¹¹⁹+2501)/35<br>1137×36⁷²⁵⁹+19<br>528×36⁶¹⁷⁸+31<br>16×36⁵⁷⁷⁹−1163<br>(36549×36⁴⁶¹⁹−289)/35<br>(979×36⁴⁵⁶³−629)/35<br>25×36³⁹³⁴−901||http://factordb.com/index.php?id=1100000002394962083&open=prime<br>http://factordb.com/index.php?id=1100000004020085177&open=prime<br>http://factordb.com/index.php?id=1100000003896952461&open=prime<br>http://factordb.com/index.php?id=1100000003807362491&open=prime<br>http://factordb.com/index.php?id=1100000003807362489&open=prime<br>http://factordb.com/index.php?id=1100000003807362488&open=prime<br>http://factordb.com/index.php?id=1100000003807362487&open=prime<br>http://factordb.com/index.php?id=1100000003807362486&open=prime<br>http://factordb.com/index.php?id=1100000003807362485&open=prime<br>http://factordb.com/index.php?id=1100000000840634476&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000002394962083&base=36<br>http://factordb.com/index.php?showid=1100000004020085177&base=36<br>http://factordb.com/index.php?showid=1100000003896952461&base=36<br>http://factordb.com/index.php?showid=1100000003807362491&base=36<br>http://factordb.com/index.php?showid=1100000003807362489&base=36<br>http://factordb.com/index.php?showid=1100000003807362488&base=36<br>http://factordb.com/index.php?showid=1100000003807362487&base=36<br>http://factordb.com/index.php?showid=1100000003807362486&base=36<br>http://factordb.com/index.php?showid=1100000003807362485&base=36<br>http://factordb.com/index.php?showid=1100000000840634476&base=36<nowiki/>||4||200000|| |} == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] smdqdjzxj6ed3um72qgt4ovms8tikmg 2819707 2819706 2026-07-28T19:41:29Z Athene241 3100061 /* Solve the problem */ 2819707 wikitext text/x-wiki {{mathematics}} '''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' > 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the "[[:w:Prime number|prime number]] [[:w:Greater than|>]] ''b''" [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, "string" is a subsequence of "meistersinger". In contrast, 758 is not a subsequence of 378259, "abc" is not a subsequence of "cbacacba", since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]]) Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|>]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''_''b'' := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' > 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s) By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] "is a subsequence of", which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}. For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. "''xy''⁺''z''" in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes > ''b'' or probable primes > ''b'' with length ≤ 100000. == Solve the problem == To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime > ''b'' in all such families. We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') "linear" families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows: * 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''. * 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by: * 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''. * 2.2. If possible, simplify the family. * 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > ''b''), and if so then remove the family from ''L''. * 3. Update ''L'', after each split examine the new families as in step 2. e.g. in decimal (base ''b'' = 10): ''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991} ''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9} and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1 and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1 221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L'' none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L'' and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed etc. Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime > ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime > ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime > ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both "numbers of primes in the set of the Athena problem in base ''b''" and "length of the largest prime in the set of the Athena problem in base ''b''" are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}. Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') * If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and the string ''xy''₁''y''₂''z'' represents a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}{''Y'' \ ''y''₂}''z''. * If ''y''₁ ∈ ''Y'' and ''y''₂ ∈ ''Y'' and ''y''₁ ≠ ''y''₂ and both the strings ''xy''₁''y''₂''z'' and ''xy''₂''y''₁''z'' represent a prime > ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime > ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''₁}''z'' ∪ ''x''{''Y'' \ ''y''₂}''z''. e.g. in decimal (base ''b'' = 10): * 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1. * 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7. * 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9. * Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1. * 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1. * 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7. * 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9. * 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}. If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''₁, ''y''₂, ..., ''y''_''n''}, then we can replace ''x''{''Y''}''z'' by ''xy''₁{''Y''}''z'' ∪ ''xy''₂{''Y''}''z'' ∪ ... ∪ ''xy''_''n''{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]]. If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list. e.g. in decimal (base ''b'' = 10): * The smallest prime in the family 5{0}27 is 5000000000000000000000000000027. * The smallest prime in the family {5}1 is 555555555551. * The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551. There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''). The following is a "[[:w:Semi-algorithm|semi-algorithm]]" that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement: # ''M'' = ''[[:w:Empty string|∅]]'' # while (''L'' ≠ ''∅'') do # choose ''x'', a shortest string in ''L'' # ''M'' := ''M'' ∪ {''x''} # ''L'' := ''L'' − ''sup''({''x''}) In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&#39;, some regular overapproximation to ''L'', until we can show ''L''&#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''₁{''L''₂}''L''₃, where each of ''L''₁, ''L''₂, ''L''₃ is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not. Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''^{''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)}), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]). To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime > ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2^''n''−2 strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length > (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'' (references: http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm), in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' < ''b'', i.e. finding the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (or prove such prime does not exist) with ''k'' < ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is < ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''^''n''+1 and ''k''×''b''^''n''−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''^''n''−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepu.txt, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379, https://pzktupel.de/Primetables/TableRepunitGen.php, https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A128164, https://oeis.org/A285642; or for prime bases ''b'': https://oeis.org/A065854, https://oeis.org/A279068), ''b''^''n''+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1) (references of this form: http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A228101), (''b''^''n''+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2) (reference of this form: http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt), (''sqrt''(''b'')×''b''^''n''+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2) (references of this form: http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/488, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://oeis.org/A084742, https://oeis.org/A084741; or for bases ''b'' with ''sqrt''(''b'') prime: https://oeis.org/A065507), ((''b''−2)×''b''^''n''+1)/(''b''−1) (''n'' ≥ 2) (reference of this form: https://oeis.org/A243404), 2×''b''^''n''+1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=6918, https://www.mersenneforum.org/showthread.php?t=19725, https://oeis.org/A119624), 2×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591), ''b''^''n''+2 (''n'' ≥ 1) (references of this form: https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067), ''b''^''n''−2 (''n'' ≥ 2) (references of this form: https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A250200, https://oeis.org/A255707, https://oeis.org/A084714; or for prime bases ''b'': https://oeis.org/A292201), (''b''−1)×''b''^''n''+1 (''n'' ≥ 1) (references of this form: http://www.noprimeleftbehind.net/Williams-primes-MP.htm, http://www.bitman.name/math/table/477, https://pzktupel.de/Primetables/TableWilliams2.php, https://oeis.org/A305531; or for prime bases ''b'': https://oeis.org/A087139), (''b''−1)×''b''^''n''−1 (''n'' ≥ 1) (references of this form: https://harvey563.tripod.com/wills.txt, http://www.noprimeleftbehind.net/Williams-primes-MM.htm, http://www.bitman.name/math/table/484, https://pzktupel.de/Primetables/TableWilliams1.php; or for prime bases ''b'': https://oeis.org/A122396), ''b''^''n''+(''b''−1) (''n'' ≥ 1) (references of this form: http://www.bitman.name/math/table/795, https://pzktupel.de/Primetables/TableWilliams6.php, https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179), ''b''^''n''−(''b''−1) (''n'' ≥ 2) (references of this form: http://www.bitman.name/math/table/792, https://pzktupel.de/Primetables/TableWilliams5.php, https://oeis.org/A113516, https://oeis.org/A343589; or for prime bases ''b'': https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435)): Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''_''n''''z'' for some ''n'' ≥ 0? Some families can be ruled out to contain no prime > ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''⁴+4×''y''⁴]]), or combine of them, e.g. * The base 9 family 2{7}: Always divisible by 2 or 5 * The base 11 family 2{5}: Always divisible by 2 or 3 * The base 14 family B{0}1: Always divisible by 3 or 5 * The base 13 family 95{0}3: Always divisible by 5, 7, or 17 * The base 16 family {4}D: Always divisible by 3, 7, or 13 * The base 16 family {8}F: Always divisible by 3, 7, or 13 * The base 21 family {7}D: Always divisible by 2, 13, or 17 * The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79 * The base 9 family {1}: Can be written as (9^''n''−1)/8 and can be factored as (3^''n''−1) × (3^''n''+1) / 8 * The base 8 family 1{0}1: Can be written as 8^''n''+1 and can be factored as (2^''n''+1) × (4^''n''−2^''n''+1) * The base 9 family 3{8}: Can be written as 4×9^''n''−1 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) * The base 16 family 1{5}: Can be written as (4×16^''n''−1)/3 and can be factored as (2×3^''n''−1) × (2×3^''n''+1) / 3 * The base 16 family {4}1: Can be written as (4×16^''n''−49)/15 and can be factored as (2×3^''n''−7) × (2×3^''n''+7) / 15 * The base 27 family 7{Q}: Can be written as 8×27^''n''−1 and can be factored as (2×3^''n''−1) × (4×9^''n''+2×3^''n''+1) * The base 27 family 9{G}: Can be written as (125×27^''n''−8)/13 and can be factored as (5×3^''n''−2) × (25×9^''n''+10×3^''n''+4) * The base 16 family {C}D: Can be written as (4×16^''n''+1)/5 and can be factored as (2×4^''n''−2×2^''n''+1) × (2×4^''n''+2×2^''n''+1) / 5 * The base 14 family 8{D}: Can be written as 9×14^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14^''n''/2−1) × (3×14^''n''/2+1) if ''n'' is even * The base 12 family {B}9B: Can be written as 12^''n''−25, it is divisible by 13 if ''n'' is odd and can be factored as (12^''n''/2−5) × (12^''n''/2+5) if ''n'' is even * The base 14 family {D}5: Can be written as 14^''n''−9, it is divisible by 5 if ''n'' is odd and can be factored as (14^''n''/2−3) × (14^''n''/2+3) if ''n'' is even * The base 17 family 1{9}: Can be written as (25×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17^''n''/2−3) × (5×17^''n''/2+3) / 16 if ''n'' is even * The base 17 family 7{9}: Can be written as (121×17^''n''−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (11×17^''n''/2−3) × (11×17^''n''/2+3) / 16 if ''n'' is even * The base 19 family 1{6}: Can be written as (4×19^''n''−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19^''n''/2−1) × (2×19^''n''/2+1) / 3 if ''n'' is even * The base 19 family 7{2}: Can be written as (64×19^''n''−1)/9, it is divisible by 5 if ''n'' is odd and can be factored as (8×19^''n''/2−1) × (8×19^''n''/2+1) / 9 if ''n'' is even * The base 24 family 3{N}: Can be written as 4×24^''n''−1, it is divisible by 5 if ''n'' is odd and can be factored as (2×24^''n''/2−1) × (2×24^''n''/2+1) if ''n'' is even By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime > ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes > ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes > ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3₅₉₂₁₉₇A, its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime, since technically, probable primality tests were used to show this (which have a ''very'' small chance of making an error, see https://t5k.org/notes/prp_prob.html) because all known primality tests run far too slowly to run on numbers of this size unless either [https://t5k.org/prove/prove3_1.html ''N''−1] or [https://t5k.org/prove/prove3_2.html ''N''+1] (or both) can be ≥ 1/3 factored). The numbers in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') are of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) for some fixed ''a'', ''b'', ''c'' such that ''a'' ≥ 1, ''b'' ≥ 2 (''b'' is the base), ''c'' ≠ 0, ''gcd''(''a'',''c'') = 1, ''gcd''(''b'',''c'') = 1. Except in the [[:w:Special case|special case]] ''c'' = ±1 and ''gcd''(''a''+''c'',''b''−1) = 1, when ''n'' is large the known [[:w:Primality test|primality test]]s for such a number are too inefficient to run. In this case one must resort to a [[:w:Probabilistic algorithm|probable]] primality test such as a [[:w:Miller–Rabin primality test|Miller–Rabin primality test]] or a [[:w:Baillie–PSW primality test|Baillie–PSW primality test]], unless a divisor of the number can be found. Since we are testing many numbers in an [[:w:Exponential growth|exponential sequence]], it is possible to use a sieving process to find divisors rather than using [[:w:Trial division|trial division]]. To do this, we made use of Geoffrey Reynolds' ''srsieve'' software (download: https://pzktupel.de/Software/srsieve_1.1.4.7z). This program uses the [[:w:Baby-step giant-step|baby-step giant-step]] [[:w:Algorithm|algorithm]] to find all primes ''p'' which divide ''a''×''b''^''n''+''c'' where ''p'' and ''n'' lie in a [[:w:Interval_(mathematics)|specified range]]. Since this program cannot handle the general case (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1 we only used it to sieve the sequence ''a''×''b''^''n''+''c'' for primes ''p'' not dividing ''gcd''(''a''+''c'',''b''−1), and initialized the list of candidates to not include ''n'' for which there is some prime ''p'' dividing ''gcd''(''a''+''c'',''b''−1) for which ''p'' dividing (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when ''a'', ''b'', and ''c'' were all odd (since then 2 divides ''a''×''b''^''n''+''c'', but 2 may not divide (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software ''LLR'' by Jean Penné. (download: http://jpenne.free.fr/index2.html). Although undocumented, it is possible to run this program on numbers of the form (''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1) when ''gcd''(''a''+''c'',''b''−1) > 1, so this program required no modifications. A script was also written which allowed one to run ''srsieve'' while ''LLR'' was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by ''LLR'' it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the "easy" bases (bases ''b'' with ≤ 150 primes > 10²⁹⁹ (base ''b'' = 26 has 83 known primes > 10²⁹⁹ and 3 unsolved families, base ''b'' = 36 has 75 known primes > 10²⁹⁹ and 4 unsolved families, base ''b'' = 17 has 99 known primes > 10²⁹⁹ and 18 unsolved families, base ''b'' = 21 has 80 known primes > 10²⁹⁹ and 12 unsolved families, base ''b'' = 19 has 201 known primes > 10²⁹⁹ and 23 unsolved families), i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed ''CM'' by Andreas Enge (download: https://www.multiprecision.org/cm/download.html), an elliptic curve primality proving implementation. == Data == These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes < 10²⁵⁰⁰⁰ for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them > 10²⁹⁹ are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others > 10²⁹⁹ are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]]) All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], "{}" means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123₄567 means 123333567 (all subscripts are written in decimal). Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11} Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111} Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221} Base 5: 22 primes (the largest of which has 96 digits (it is 10₉₃13, and its algebraic form is 5⁹⁵+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013} Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} Base 7: 71 primes (the largest of which has 17 digits (it is 3₁₆1, and its algebraic form is (7¹⁷−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331} Base 8: 75 primes (the largest of which has 221 digits (it is 4₂₂₀7, and its algebraic form is (4×8²²¹+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447} Base 9: 151 primes (the largest of which has 1161 digits (it is 30₁₁₅₈11, and its algebraic form is 3×9¹¹⁶⁰+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011} Base 10: 77 primes (the largest of which has 31 digits (it is 50₂₈27, and its algebraic form is 5×10³⁰+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} Base 11: 1068 primes (including 1 unproven probable prime: 57₆₂₆₆₈), the largest of which has 62669 digits (it is 57₆₂₆₆₈, and its algebraic form is (57×11⁶²⁶⁶⁸−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11] Base 12: 106 primes (the largest of which has 42 digits (it is 40₃₉77, and its algebraic form is 4×12⁴¹+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077} Base 13: 3197 primes (including 4 unproven probable primes: C5₂₃₇₅₅C, 80₃₂₀₁₇111, 95₁₉₇₄₂₀, A3₅₉₂₁₉₇A), the largest of which has 592199 digits (it is A3₅₉₂₁₉₇A, and its algebraic form is (41×13⁵⁹²¹⁹⁸+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13] Base 14: 650 primes, the largest of which has 19699 digits (it is 4D₁₉₆₉₈, and its algebraic form is 5×14¹⁹⁶⁹⁸−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14] Base 15: 1284 primes, the largest of which has 157 digits (it is 7₁₅₅97, and its algebraic form is (15¹⁵⁷+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15] Base 16: 2347 primes (including 3 unproven probable primes: DB₃₂₂₃₄, 4₇₂₇₈₅DD, 3₁₁₆₁₃₇AF), the largest of which has 116139 digits (it is 3₁₁₆₁₃₇AF, and its algebraic form is (16¹¹⁶¹³⁹+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16] Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] Base 18: 549 primes, the largest of which has 6271 digits (it is C0₆₂₆₈C5, and its algebraic form is 12×18⁶²⁷⁰+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18] Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] Base 20: 3314 primes, the largest of which has 6271 digits (it is G0₆₂₆₉D, and its algebraic form is 16×20⁶²⁷⁰+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20] Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] Base 22: 8003 primes (including 1 unproven probable prime: BK₂₂₀₀₁5), the largest of which has 22003 digits (it is BK₂₂₀₀₁5, and its algebraic form is (251×22²²⁰⁰²−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22] Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23] Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N₈₁₂₉LN, and its algebraic form is 13249×24⁸¹³¹−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24] Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25] Base 26: 25256 known primes (including 7 unproven probable primes: 5₁₉₃₉₁6F, 7₂₀₂₇₉OL, LD0₂₀₉₇₅7, 6K₂₃₃₀₀5, J0₄₄₃₀₃KCB, M0₆₁₁₈₆2BB, 85M₁₉₇₀₆₀B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26] Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27] Base 28: 25528 known primes (including 3 unproven probable primes: N6₂₄₀₅₁LR, 5OA₃₁₂₃₈F, O4O₉₄₅₃₅9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28] Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29] Base 30: 2619 primes (including 1 unproven probable prime: I0₂₄₆₀₈D), the largest of which has 34206 digits (it is OT₃₄₂₀₅, and its algebraic form is 25×30³⁴²⁰⁵−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30] Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31] Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32] Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33] Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34] Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35] Base 36: 35286 known primes (including 3 unproven probable primes: 7K₂₆₅₆₇Z, S0₇₅₀₀₇8H, P₈₁₉₉₃SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36] == Condensed table for bases 2 ≤ ''b'' ≤ 36 == {|class="wikitable" ||''b''||number of primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||base-''b'' form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' (write "''d''_''n''" if there are 5 or more (''n'') consecutive same digits ''d'')||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||length of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' in decimal||algebraic ((''a''×''b''^''n''+''c'')/''gcd''(''a''+''c'',''b''−1)) form of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||''factordb'' entry of the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b''||the top 10 known primes (or probable prime, which is a minimal prime assuming its primality) in the Athena problem base ''b'' written in base ''b'' (use lower case letters instead of upper case letters)||number of left families in the Athena problem base ''b'' (all of these left families are linear families)||searching limit of length for the left families in the Athena problem base ''b'' (if there are different searching limits for the left families in base ''b'', choose the lowest searching limit)|| |- ||2||1||11||2||1||3||http://factordb.com/index.php?id=3&open=ecm||http://factordb.com/index.php?showid=3&base=2||0||–|| |- ||3||3||111<br>21<br>12||3<br>2<br>2||2<br>1<br>1||13<br>7<br>5||http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=13&base=3<br>http://factordb.com/index.php?showid=7&base=3<br>http://factordb.com/index.php?showid=5&base=3<nowiki/>||0||–|| |- ||4||5||221<br>31<br>23<br>13<br>11||3<br>2<br>2<br>2<br>2||2<br>2<br>2<br>1<br>1||41<br>13<br>11<br>7<br>5||http://factordb.com/index.php?id=41&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<br>http://factordb.com/index.php?id=7&open=ecm<br>http://factordb.com/index.php?id=5&open=ecm<nowiki/>||http://factordb.com/index.php?showid=41&base=4<br>http://factordb.com/index.php?showid=13&base=4<br>http://factordb.com/index.php?showid=11&base=4<br>http://factordb.com/index.php?showid=7&base=4<br>http://factordb.com/index.php?showid=5&base=4<nowiki/>||0||–|| |- ||5||22||10₉₃13<br>300031<br>44441<br>33331<br>33001<br>30301<br>14444<br>10103<br>3101<br>414||96<br>6<br>5<br>5<br>5<br>5<br>5<br>5<br>4<br>3||67<br>4<br>4<br>4<br>4<br>4<br>4<br>3<br>3<br>3||5⁹⁵+8<br>9391<br>3121<br>2341<br>2251<br>1951<br>1249<br>653<br>401<br>109||http://factordb.com/index.php?id=1100000000034686071&open=ecm<br>http://factordb.com/index.php?id=9391&open=ecm<br>http://factordb.com/index.php?id=3121&open=ecm<br>http://factordb.com/index.php?id=2341&open=ecm<br>http://factordb.com/index.php?id=2251&open=ecm<br>http://factordb.com/index.php?id=1951&open=ecm<br>http://factordb.com/index.php?id=1249&open=ecm<br>http://factordb.com/index.php?id=653&open=ecm<br>http://factordb.com/index.php?id=401&open=ecm<br>http://factordb.com/index.php?id=109&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000034686071&base=5<br>http://factordb.com/index.php?showid=9391&base=5<br>http://factordb.com/index.php?showid=3121&base=5<br>http://factordb.com/index.php?showid=2341&base=5<br>http://factordb.com/index.php?showid=2251&base=5<br>http://factordb.com/index.php?showid=1951&base=5<br>http://factordb.com/index.php?showid=1249&base=5<br>http://factordb.com/index.php?showid=653&base=5<br>http://factordb.com/index.php?showid=401&base=5<br>http://factordb.com/index.php?showid=109&base=5<nowiki/>||0||–|| |- ||6||11||40041<br>4441<br>4401<br>51<br>45<br>35<br>31<br>25<br>21<br>15||5<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||4<br>4<br>4<br>2<br>2<br>2<br>2<br>2<br>2<br>2||5209<br>1033<br>1009<br>31<br>29<br>23<br>19<br>17<br>13<br>11||http://factordb.com/index.php?id=5209&open=ecm<br>http://factordb.com/index.php?id=1033&open=ecm<br>http://factordb.com/index.php?id=1009&open=ecm<br>http://factordb.com/index.php?id=31&open=ecm<br>http://factordb.com/index.php?id=29&open=ecm<br>http://factordb.com/index.php?id=23&open=ecm<br>http://factordb.com/index.php?id=19&open=ecm<br>http://factordb.com/index.php?id=17&open=ecm<br>http://factordb.com/index.php?id=13&open=ecm<br>http://factordb.com/index.php?id=11&open=ecm<nowiki/>||http://factordb.com/index.php?showid=5209&base=6<br>http://factordb.com/index.php?showid=1033&base=6<br>http://factordb.com/index.php?showid=1009&base=6<br>http://factordb.com/index.php?showid=31&base=6<br>http://factordb.com/index.php?showid=29&base=6<br>http://factordb.com/index.php?showid=23&base=6<br>http://factordb.com/index.php?showid=19&base=6<br>http://factordb.com/index.php?showid=17&base=6<br>http://factordb.com/index.php?showid=13&base=6<br>http://factordb.com/index.php?showid=11&base=6<nowiki/>||0||–|| |- ||7||71||3₁₆1<br>510₇1<br>3₆01<br>1100021<br>531101<br>351101<br>300053<br>150001<br>100121<br>40054||17<br>10<br>8<br>7<br>6<br>6<br>6<br>6<br>6<br>5||15<br>9<br>7<br>6<br>5<br>5<br>5<br>5<br>5<br>4||(7¹⁷−5)/2<br>36×7⁸+1<br>(7⁸−47)/2<br>134471<br>91631<br>62819<br>50459<br>28813<br>16871<br>9643||http://factordb.com/index.php?id=116315256993601&open=ecm<br>http://factordb.com/index.php?id=207532837&open=ecm<br>http://factordb.com/index.php?id=2882377&open=ecm<br>http://factordb.com/index.php?id=134471&open=ecm<br>http://factordb.com/index.php?id=91631&open=ecm<br>http://factordb.com/index.php?id=62819&open=ecm<br>http://factordb.com/index.php?id=50459&open=ecm<br>http://factordb.com/index.php?id=28813&open=ecm<br>http://factordb.com/index.php?id=16871&open=ecm<br>http://factordb.com/index.php?id=9643&open=ecm<nowiki/>||http://factordb.com/index.php?showid=116315256993601&base=7<br>http://factordb.com/index.php?showid=207532837&base=7<br>http://factordb.com/index.php?showid=2882377&base=7<br>http://factordb.com/index.php?showid=134471&base=7<br>http://factordb.com/index.php?showid=91631&base=7<br>http://factordb.com/index.php?showid=62819&base=7<br>http://factordb.com/index.php?showid=50459&base=7<br>http://factordb.com/index.php?showid=28813&base=7<br>http://factordb.com/index.php?showid=16871&base=7<br>http://factordb.com/index.php?showid=9643&base=7<nowiki/>||0||–|| |- ||8||75||4₂₂₀7<br>5₁₃25<br>7₁₂1<br>77774₆1<br>74₇1<br>4₈1<br>5₅025<br>5550525<br>5500525<br>4₅77||221<br>15<br>13<br>11<br>9<br>9<br>8<br>7<br>7<br>7||200<br>14<br>12<br>10<br>9<br>8<br>8<br>7<br>7<br>7||(4×8²²¹+17)/7<br>(5×8¹⁵−173)/7<br>8¹³−7<br>(28669×8⁷−25)/7<br>(53×8⁸−25)/7<br>(4×8⁹−25)/7<br>(5×8⁸−2413)/7<br>1495381<br>1474901<br>(4×8⁷+185)/7||http://factordb.com/index.php?id=1100000000416605822&open=ecm<br>http://factordb.com/index.php?id=25131694349141&open=ecm<br>http://factordb.com/index.php?id=549755813881&open=ecm<br>http://factordb.com/index.php?id=8589035809&open=ecm<br>http://factordb.com/index.php?id=127027489&open=ecm<br>http://factordb.com/index.php?id=76695841&open=ecm<br>http://factordb.com/index.php?id=11983381&open=ecm<br>http://factordb.com/index.php?id=1495381&open=ecm<br>http://factordb.com/index.php?id=1474901&open=ecm<br>http://factordb.com/index.php?id=1198399&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000416605822&base=8<br>http://factordb.com/index.php?showid=25131694349141&base=8<br>http://factordb.com/index.php?showid=549755813881&base=8<br>http://factordb.com/index.php?showid=8589035809&base=8<br>http://factordb.com/index.php?showid=127027489&base=8<br>http://factordb.com/index.php?showid=76695841&base=8<br>http://factordb.com/index.php?showid=11983381&base=8<br>http://factordb.com/index.php?showid=1495381&base=8<br>http://factordb.com/index.php?showid=1474901&base=8<br>http://factordb.com/index.php?showid=1198399&base=8<nowiki/>||0||–|| |- ||9||151||30₁₁₅₈11<br>27₆₈₆07<br>76₃₂₉2<br>561₃₆<br>10₂₅57<br>30₂₀51<br>8₁₉335<br>727₁₅07<br>51₁₃61<br>10₁₁507||1161<br>689<br>331<br>38<br>28<br>23<br>22<br>19<br>16<br>15||1108<br>657<br>316<br>37<br>26<br>22<br>21<br>19<br>16<br>14||3×9¹¹⁶⁰+10<br>(23×9⁶⁸⁸−511)/8<br>(31×9³³⁰−19)/4<br>(409×9³⁶−1)/8<br>9²⁷+52<br>3×9²²+46<br>9²²−454<br>(527×9¹⁷−511)/8<br>(41×9¹⁵+359)/8<br>9¹⁴+412||http://factordb.com/index.php?id=1100000002376318423&open=prime<br>http://factordb.com/index.php?id=1100000002495467486&open=prime<br>http://factordb.com/index.php?id=1100000002359003642&open=prime<br>http://factordb.com/index.php?id=1100000001554010824&open=ecm<br>http://factordb.com/index.php?id=1100000002512830927&open=ecm<br>http://factordb.com/index.php?id=1100000000032261811&open=ecm<br>http://factordb.com/index.php?id=1100000002495736583&open=ecm<br>http://factordb.com/index.php?id=1100000003446800389&open=ecm<br>http://factordb.com/index.php?id=1055192051985121&open=ecm<br>http://factordb.com/index.php?id=22876792455373&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002376318423&base=9<br>http://factordb.com/index.php?showid=1100000002495467486&base=9<br>http://factordb.com/index.php?showid=1100000002359003642&base=9<br>http://factordb.com/index.php?showid=1100000001554010824&base=9<br>http://factordb.com/index.php?showid=1100000002512830927&base=9<br>http://factordb.com/index.php?showid=1100000000032261811&base=9<br>http://factordb.com/index.php?showid=1100000002495736583&base=9<br>http://factordb.com/index.php?showid=1100000003446800389&base=9<br>http://factordb.com/index.php?showid=1055192051985121&base=9<br>http://factordb.com/index.php?showid=22876792455373&base=9<nowiki/>||0||–|| |- ||10||77||50₂₈27<br>5₁₁1<br>805₅1<br>66600049<br>66000049<br>60₅49<br>220₅1<br>5200007<br>946669<br>666649||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||31<br>12<br>8<br>8<br>8<br>8<br>8<br>7<br>6<br>6||5×10³⁰+27<br>(5×10¹²−41)/9<br>(725×10⁶−41)/9<br>66600049<br>66000049<br>6×10⁷+49<br>22×10⁶+1<br>5200007<br>946669<br>666649||http://factordb.com/index.php?id=1100000000204142046&open=ecm<br>http://factordb.com/index.php?id=555555555551&open=ecm<br>http://factordb.com/index.php?id=80555551&open=ecm<br>http://factordb.com/index.php?id=66600049&open=ecm<br>http://factordb.com/index.php?id=66000049&open=ecm<br>http://factordb.com/index.php?id=60000049&open=ecm<br>http://factordb.com/index.php?id=22000001&open=ecm<br>http://factordb.com/index.php?id=5200007&open=ecm<br>http://factordb.com/index.php?id=946669&open=ecm<br>http://factordb.com/index.php?id=666649&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000204142046&base=10<br>http://factordb.com/index.php?showid=555555555551&base=10<br>http://factordb.com/index.php?showid=80555551&base=10<br>http://factordb.com/index.php?showid=66600049&base=10<br>http://factordb.com/index.php?showid=66000049&base=10<br>http://factordb.com/index.php?showid=60000049&base=10<br>http://factordb.com/index.php?showid=22000001&base=10<br>http://factordb.com/index.php?showid=5200007&base=10<br>http://factordb.com/index.php?showid=946669&base=10<br>http://factordb.com/index.php?showid=666649&base=10<nowiki/>||0||–|| |- ||11||1068||57₆₂₆₆₈<br>557₁₀₁₁<br>7₇₅₉44<br>A₇₁₃58<br>85₂₂₀05<br>507₂₀₆<br>5₁₆₁2A<br>50₁₂₆57<br>10₁₂₅51<br>326₁₂₂||62669<br>1013<br>761<br>715<br>223<br>208<br>163<br>129<br>128<br>124||65263<br>1055<br>793<br>745<br>233<br>217<br>170<br>134<br>133<br>129||(57×11⁶²⁶⁶⁸−7)/10<br>(607×11¹⁰¹¹−7)/10<br>(7×11⁷⁶¹−367)/10<br>11⁷¹⁵−58<br>(17×11²²²−111)/2<br>(557×11²⁰⁶−7)/10<br>(11¹⁶³−57)/2<br>5×11¹²⁸+62<br>11¹²⁷+56<br>(178×11¹²²−3)/5||http://factordb.com/index.php?id=1100000003573679860&open=prime<br>http://factordb.com/index.php?id=1100000002361376522&open=prime<br>http://factordb.com/index.php?id=1100000002505568840&open=prime<br>http://factordb.com/index.php?id=1100000003576826487&open=prime<br>http://factordb.com/index.php?id=1100000003576826769&open=ecm<br>http://factordb.com/index.php?id=1100000002518512744&open=ecm<br>http://factordb.com/index.php?id=1100000002391585327&open=ecm<br>http://factordb.com/index.php?id=1100000002632393378&open=ecm<br>http://factordb.com/index.php?id=1100000002391531300&open=ecm<br>http://factordb.com/index.php?id=1100000003576826781&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003573679860&base=11<br>http://factordb.com/index.php?showid=1100000002361376522&base=11<br>http://factordb.com/index.php?showid=1100000002505568840&base=11<br>http://factordb.com/index.php?showid=1100000003576826487&base=11<br>http://factordb.com/index.php?showid=1100000003576826769&base=11<br>http://factordb.com/index.php?showid=1100000002518512744&base=11<br>http://factordb.com/index.php?showid=1100000002391585327&base=11<br>http://factordb.com/index.php?showid=1100000002632393378&base=11<br>http://factordb.com/index.php?showid=1100000002391531300&base=11<br>http://factordb.com/index.php?showid=1100000003576826781&base=11<nowiki/>||0||–|| |- ||12||106||40₃₉77<br>B0₂₇9B<br>B₆99B<br>AA0₅1<br>B00099B<br>AAA0001<br>BBBAA1<br>A00065<br>44AAA1<br>BBBB1||42<br>30<br>9<br>8<br>7<br>7<br>6<br>6<br>6<br>5||45<br>33<br>10<br>9<br>8<br>8<br>7<br>7<br>7<br>6||4×12⁴¹+91<br>11×12²⁹+119<br>12⁹−313<br>130×12⁶+1<br>32847239<br>32555521<br>2985817<br>2488397<br>1097113<br>248821||http://factordb.com/index.php?id=1100000002375054575&open=ecm<br>http://factordb.com/index.php?id=1100000002354113100&open=ecm<br>http://factordb.com/index.php?id=5159780039&open=ecm<br>http://factordb.com/index.php?id=388177921&open=ecm<br>http://factordb.com/index.php?id=32847239&open=ecm<br>http://factordb.com/index.php?id=32555521&open=ecm<br>http://factordb.com/index.php?id=2985817&open=ecm<br>http://factordb.com/index.php?id=2488397&open=ecm<br>http://factordb.com/index.php?id=1097113&open=ecm<br>http://factordb.com/index.php?id=248821&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002375054575&base=12<br>http://factordb.com/index.php?showid=1100000002354113100&base=12<br>http://factordb.com/index.php?showid=5159780039&base=12<br>http://factordb.com/index.php?showid=388177921&base=12<br>http://factordb.com/index.php?showid=32847239&base=12<br>http://factordb.com/index.php?showid=32555521&base=12<br>http://factordb.com/index.php?showid=2985817&base=12<br>http://factordb.com/index.php?showid=2488397&base=12<br>http://factordb.com/index.php?showid=1097113&base=12<br>http://factordb.com/index.php?showid=248821&base=12<nowiki/>||0||–|| |- ||13||3197||A3₅₉₂₁₉₇A<br>95₁₉₇₄₂₀<br>80₃₂₀₁₇111<br>C5₂₃₇₅₅C<br>C₁₀₆₃₁92<br>B0₆₅₄₀BBA<br>390₆₂₆₆1<br>1770₂₇₀₃17<br>720₂₂₉₇2<br>930₁₅₅₁1||592199<br>197421<br>32021<br>23757<br>10633<br>6544<br>6269<br>2708<br>2300<br>1554||659677<br>219916<br>35670<br>26464<br>11845<br>7290<br>6983<br>3016<br>2562<br>1731||(41×13⁵⁹²¹⁹⁸+27)/4<br>(113×13¹⁹⁷⁴²⁰−5)/12<br>8×13³²⁰²⁰+183<br>(149×13²³⁷⁵⁶+79)/12<br>13¹⁰⁶³³−50<br>11×13⁶⁵⁴³+2012<br>48×13⁶²⁶⁷+1<br>267×13²⁷⁰⁵+20<br>93×13²²⁹⁸+2<br>120×13¹⁵⁵²+1||http://factordb.com/index.php?id=1100000005489162806&open=prime<br>http://factordb.com/index.php?id=1100000003943359311&open=prime<br>http://factordb.com/index.php?id=1100000000490878060&open=prime<br>http://factordb.com/index.php?id=1100000003590647776&open=prime<br>http://factordb.com/index.php?id=1100000003590493750&open=prime<br>http://factordb.com/index.php?id=1100000002616382906&open=prime<br>http://factordb.com/index.php?id=1100000000765961441&open=prime<br>http://factordb.com/index.php?id=1100000003590430825&open=prime<br>http://factordb.com/index.php?id=1100000002632396910&open=prime<br>http://factordb.com/index.php?id=1100000000765961452&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005489162806&base=13<br>http://factordb.com/index.php?showid=1100000003943359311&base=13<br>http://factordb.com/index.php?showid=1100000000490878060&base=13<br>http://factordb.com/index.php?showid=1100000003590647776&base=13<br>http://factordb.com/index.php?showid=1100000003590493750&base=13<br>http://factordb.com/index.php?showid=1100000002616382906&base=13<br>http://factordb.com/index.php?showid=1100000000765961441&base=13<br>http://factordb.com/index.php?showid=1100000003590430825&base=13<br>http://factordb.com/index.php?showid=1100000002632396910&base=13<br>http://factordb.com/index.php?showid=1100000000765961452&base=13<nowiki/>||0||–|| |- ||14||650||4D₁₉₆₉₈<br>34D₇₀₈<br>8D₁₄₁85<br>8₈₆B<br>40₈₃49<br>8C₇₉3<br>18₇₉B<br>6B₇₇2B<br>4₆₃09<br>A₅₉3||19699<br>710<br>144<br>87<br>86<br>81<br>81<br>80<br>65<br>60||22578<br>814<br>165<br>100<br>99<br>93<br>92<br>92<br>74<br>69||5×14¹⁹⁶⁹⁸−1<br>47×14⁷⁰⁸−1<br>9×14¹⁴³−79<br>(8×14⁸⁷+31)/13<br>4×14⁸⁵+65<br>(116×14⁸⁰−129)/13<br>(21×14⁸⁰+31)/13<br>(89×14⁷⁹−1649)/13<br>(4×14⁶⁵−667)/13<br>(10×14⁶⁰−101)/13||http://factordb.com/index.php?id=1100000000884560233&open=prime<br>http://factordb.com/index.php?id=1100000001540144903&open=prime<br>http://factordb.com/index.php?id=1100000003575856650&open=ecm<br>http://factordb.com/index.php?id=1100000002321014379&open=ecm<br>http://factordb.com/index.php?id=1100000000823937973&open=ecm<br>http://factordb.com/index.php?id=1100000002631073246&open=ecm<br>http://factordb.com/index.php?id=1100000002384401372&open=ecm<br>http://factordb.com/index.php?id=1100000002631077787&open=ecm<br>http://factordb.com/index.php?id=1100000000840126683&open=ecm<br>http://factordb.com/index.php?id=1100000002321038522&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000884560233&base=14<br>http://factordb.com/index.php?showid=1100000001540144903&base=14<br>http://factordb.com/index.php?showid=1100000003575856650&base=14<br>http://factordb.com/index.php?showid=1100000002321014379&base=14<br>http://factordb.com/index.php?showid=1100000000823937973&base=14<br>http://factordb.com/index.php?showid=1100000002631073246&base=14<br>http://factordb.com/index.php?showid=1100000002384401372&base=14<br>http://factordb.com/index.php?showid=1100000002631077787&base=14<br>http://factordb.com/index.php?showid=1100000000840126683&base=14<br>http://factordb.com/index.php?showid=1100000002321038522&base=14<nowiki/>||0||–|| |- ||15||1284||7₁₅₅97<br>E₁₄₅397<br>96₁₀₄08<br>7₇₃CE<br>7₅₉CCE<br>50₃₃17<br>EB₃₁<br>6330₂₆1<br>7050₂₄B<br>B70₂₄1||157<br>148<br>107<br>75<br>62<br>36<br>32<br>30<br>28<br>27||185<br>175<br>126<br>88<br>73<br>42<br>38<br>35<br>33<br>32||(15¹⁵⁷+59)/2<br>15¹⁴⁸−2558<br>(66×15¹⁰⁶−619)/7<br>(15⁷⁵+163)/2<br>(15⁶²+2413)/2<br>5×15³⁵+22<br>(207×15³¹−11)/14<br>1398×15²⁷+1<br>1580×15²⁵+11<br>172×15²⁵+1||http://factordb.com/index.php?id=1100000002454891840&open=ecm<br>http://factordb.com/index.php?id=1100000002454900849&open=ecm<br>http://factordb.com/index.php?id=1100000000823937997&open=ecm<br>http://factordb.com/index.php?id=1100000003588407143&open=ecm<br>http://factordb.com/index.php?id=1100000003588407386&open=ecm<br>http://factordb.com/index.php?id=1100000002632398579&open=ecm<br>http://factordb.com/index.php?id=1100000002321033312&open=ecm<br>http://factordb.com/index.php?id=1100000002391199877&open=ecm<br>http://factordb.com/index.php?id=1100000003588407806&open=ecm<br>http://factordb.com/index.php?id=1100000000851967288&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000002454891840&base=15<br>http://factordb.com/index.php?showid=1100000002454900849&base=15<br>http://factordb.com/index.php?showid=1100000000823937997&base=15<br>http://factordb.com/index.php?showid=1100000003588407143&base=15<br>http://factordb.com/index.php?showid=1100000003588407386&base=15<br>http://factordb.com/index.php?showid=1100000002632398579&base=15<br>http://factordb.com/index.php?showid=1100000002321033312&base=15<br>http://factordb.com/index.php?showid=1100000002391199877&base=15<br>http://factordb.com/index.php?showid=1100000003588407806&base=15<br>http://factordb.com/index.php?showid=1100000000851967288&base=15<nowiki/>||0||–|| |- ||16||2347||3₁₁₆₁₃₇AF<br>4₇₂₇₈₅DD<br>DB₃₂₂₃₄<br>D0B₁₇₈₀₄<br>5BC₃₇₀₀D<br>90₃₅₄₂91<br>300F₁₉₆₀AF<br>20₁₇₁₃321<br>F8₁₅₁₇F<br>FAF₁₀₆₂45||116139<br>72787<br>32235<br>17806<br>3703<br>3545<br>1965<br>1717<br>1519<br>1066||139845<br>87644<br>38815<br>21441<br>4459<br>4269<br>2366<br>2067<br>1830<br>1284||(16¹¹⁶¹³⁹+619)/5<br>(4×16⁷²⁷⁸⁷+2291)/15<br>(206×16³²²³⁴−11)/15<br>(3131×16¹⁷⁸⁰⁴−11)/15<br>(459×16³⁷⁰¹+1)/5<br>9×16³⁵⁴⁴+145<br>769×16¹⁹⁶²−81<br>2×16¹⁷¹⁶+801<br>(233×16¹⁵¹⁸+97)/15<br>251×16¹⁰⁶⁴−187||http://factordb.com/index.php?id=1100000003851731988&open=prime<br>http://factordb.com/index.php?id=1100000003615909841&open=prime<br>http://factordb.com/index.php?id=1100000002383583629&open=prime<br>http://factordb.com/index.php?id=1100000003589278511&open=prime<br>http://factordb.com/index.php?id=1100000000993764322&open=prime<br>http://factordb.com/index.php?id=1100000000633424191&open=prime<br>http://factordb.com/index.php?id=1100000003588368750&open=prime<br>http://factordb.com/index.php?id=1100000003588386735&open=prime<br>http://factordb.com/index.php?id=1100000000633744824&open=prime<br>http://factordb.com/index.php?id=1100000003588387610&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003851731988&base=16<br>http://factordb.com/index.php?showid=1100000003615909841&base=16<br>http://factordb.com/index.php?showid=1100000002383583629&base=16<br>http://factordb.com/index.php?showid=1100000003589278511&base=16<br>http://factordb.com/index.php?showid=1100000000993764322&base=16<br>http://factordb.com/index.php?showid=1100000000633424191&base=16<br>http://factordb.com/index.php?showid=1100000003588368750&base=16<br>http://factordb.com/index.php?showid=1100000003588386735&base=16<br>http://factordb.com/index.php?showid=1100000000633744824&base=16<br>http://factordb.com/index.php?showid=1100000003588387610&base=16<nowiki/>||0||–|| |- ||17||10415~10427||95F₁₉₈₈₅₅<br>B0₁₈₉₀₈₃DB<br>F70₁₈₆₇₆₇1<br>970₁₆₆₀₄₇1<br>510₁₂₄₀₇₄D<br>49₁₁₁₃₃₃<br>B₆₇₁₀₃2E<br>570₅₁₃₁₀1<br>E9B₄₄₇₃₂<br>D0GD₃₇₀₉₆||198857<br>189086<br>186770<br>166050<br>124077<br>111334<br>67105<br>51313<br>44734<br>37099||244684<br>232661<br>229811<br>204316<br>152670<br>136991<br>82570<br>63138<br>55043<br>45649||(2543×17¹⁹⁸⁸⁵⁵−15)/16<br>11×17¹⁸⁹⁰⁸⁵+232<br>262×17¹⁸⁶⁷⁶⁸+1<br>160×17¹⁶⁶⁰⁴⁸+1<br>86×17¹²⁴⁰⁷⁵+13<br>(73×17¹¹¹³³³−9)/16<br>(11×17⁶⁷¹⁰⁵−2411)/16<br>92×17⁵¹³¹¹+1<br>(3963×17⁴⁴⁷³²−11)/16<br>(60381×17³⁷⁰⁹⁶−13)/16||http://factordb.com/index.php?id=1100000008610514108&open=prime<br>http://factordb.com/index.php?id=1100000008610515753&open=prime<br>http://factordb.com/index.php?id=1100000000765961429&open=prime<br>http://factordb.com/index.php?id=1100000000765961411&open=prime<br>http://factordb.com/index.php?id=1100000008610516879&open=prime<br>http://factordb.com/index.php?id=1100000000808118219&open=prime<br>http://factordb.com/index.php?id=1100000003993647842&open=prime<br>http://factordb.com/index.php?id=1100000000765961389&open=prime<br>http://factordb.com/index.php?id=1100000003883765450&open=prime<br>http://factordb.com/index.php?id=1100000003848346668&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008610514108&base=17<br>http://factordb.com/index.php?showid=1100000008610515753&base=17<br>http://factordb.com/index.php?showid=1100000000765961429&base=17<br>http://factordb.com/index.php?showid=1100000000765961411&base=17<br>http://factordb.com/index.php?showid=1100000008610516879&base=17<br>http://factordb.com/index.php?showid=1100000000808118219&base=17<br>http://factordb.com/index.php?showid=1100000003993647842&base=17<br>http://factordb.com/index.php?showid=1100000000765961389&base=17<br>http://factordb.com/index.php?showid=1100000003883765450&base=17<br>http://factordb.com/index.php?showid=1100000003848346668&base=17<nowiki/>||12||200000|| |- ||18||549||C0₆₂₆₈C5<br>H₇₆₆FH<br>80₂₉₈B<br>C0₁₁₆F5<br>HD₉₃<br>GG0₃₀1<br>CF₃₀5<br>B₁₉6B<br>CCF₁₄5<br>7₁₄G7||6271<br>768<br>300<br>119<br>94<br>33<br>32<br>21<br>17<br>16||7872<br>965<br>377<br>150<br>118<br>42<br>41<br>27<br>22<br>20||12×18⁶²⁷⁰+221<br>18⁷⁶⁸−37<br>8×18²⁹⁹+11<br>12×18¹¹⁸+275<br>(302×18⁹³−13)/17<br>304×18³¹+1<br>(219×18³¹−185)/17<br>(11×18²¹−1541)/17<br>(3891×18¹⁵−185)/17<br>(7×18¹⁶+2747)/17||http://factordb.com/index.php?id=1100000003590442437&open=prime<br>http://factordb.com/index.php?id=1100000003590430490&open=prime<br>http://factordb.com/index.php?id=1100000002355574745&open=prime<br>http://factordb.com/index.php?id=1100000002632837015&open=ecm<br>http://factordb.com/index.php?id=1100000002321052894&open=ecm<br>http://factordb.com/index.php?id=1100000000819230161&open=ecm<br>http://factordb.com/index.php?id=1100000002631240657&open=ecm<br>http://factordb.com/index.php?id=1100000003590430474&open=ecm<br>http://factordb.com/index.php?id=1100000003590430470&open=ecm<br>http://factordb.com/index.php?id=1100000003590430465&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000003590442437&base=18<br>http://factordb.com/index.php?showid=1100000003590430490&base=18<br>http://factordb.com/index.php?showid=1100000002355574745&base=18<br>http://factordb.com/index.php?showid=1100000002632837015&base=18<br>http://factordb.com/index.php?showid=1100000002321052894&base=18<br>http://factordb.com/index.php?showid=1100000000819230161&base=18<br>http://factordb.com/index.php?showid=1100000002631240657&base=18<br>http://factordb.com/index.php?showid=1100000003590430474&base=18<br>http://factordb.com/index.php?showid=1100000003590430470&base=18<br>http://factordb.com/index.php?showid=1100000003590430465&base=18<nowiki/>||0||–|| |- ||19||31417~31434||1E70₁₂₂₈₉₆1<br>40₁₂₁₈₄₆HB5<br>35₁₂₀₅₆₂<br>FH0H₁₁₂₆₅₉<br>FG6₁₁₀₉₈₄<br>H₈₆₂₉₁6<br>D90₇₃₀₄₆9<br>4F0₄₉₈₄₇6<br>2₄₈₂₂₄7<br>2₄₅₈₈₆7A||122900<br>121850<br>120563<br>112662<br>110986<br>86292<br>73049<br>49850<br>48225<br>45888||157158<br>155816<br>154170<br>144067<br>110347<br>141924<br>93412<br>63746<br>61667<br>58679||634×19¹²²⁸⁹⁷+1<br>4×19¹²¹⁸⁴⁹+6351<br>(59×19¹²⁰⁵⁶²−5)/18<br>(103301×19¹¹²⁶⁵⁹−17)/18<br>(904×19¹¹⁰⁹⁸⁴−1)/3<br>(17×19⁸⁶²⁹²−215)/18<br>256×19⁷³⁰⁴⁷+9<br>91×19⁴⁹⁸⁴⁸+6<br>(19⁴⁸²²⁵+44)/9<br>(19⁴⁵⁸⁸⁸+926)/9||http://factordb.com/index.php?id=1100000001582289581&open=prime<br>http://factordb.com/index.php?id=1100000008755307222&open=prime<br>http://factordb.com/index.php?id=1100000005513825027&open=prime<br>http://factordb.com/index.php?id=1100000008755311453&open=prime<br>http://factordb.com/index.php?id=1100000000808118212&open=prime<br>http://factordb.com/index.php?id=1100000004163040839&open=prime<br>http://factordb.com/index.php?id=1100000003998413751&open=prime<br>http://factordb.com/index.php?id=1100000000808118332&open=prime<br>http://factordb.com/index.php?id=1100000003949188041&open=prime<br>http://factordb.com/index.php?id=1100000003949189035&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000001582289581&base=19<br>http://factordb.com/index.php?showid=1100000008755307222&base=19<br>http://factordb.com/index.php?showid=1100000005513825027&base=19<br>http://factordb.com/index.php?showid=1100000008755311453&base=19<br>http://factordb.com/index.php?showid=1100000000808118212&base=19<br>http://factordb.com/index.php?showid=1100000004163040839&base=19<br>http://factordb.com/index.php?showid=1100000003998413751&base=19<br>http://factordb.com/index.php?showid=1100000000808118332&base=19<br>http://factordb.com/index.php?showid=1100000003949188041&base=19<br>http://factordb.com/index.php?showid=1100000003949189035&base=19<nowiki/>||17||200000|| |- ||20||3314||G0₆₂₆₉D<br>CD₂₄₄₉<br>50₁₁₆₃AJ<br>J₆₅₅05J<br>JCJ₆₂₉<br>E₅₆₆C7<br>3A₅₂₇3<br>G₄₄₇99<br>EC0₄₂₉7<br>40₃₈₇404B||6271<br>2450<br>1166<br>658<br>631<br>568<br>529<br>449<br>432<br>392||8159<br>3188<br>1517<br>857<br>821<br>739<br>688<br>585<br>562<br>510||16×20⁶²⁷⁰+13<br>(241×20²⁴⁴⁹−13)/19<br>5×20¹¹⁶⁵+219<br>20⁶⁵⁸−7881<br>393×20⁶²⁹−1<br>(14×20⁵⁶⁸−907)/19<br>(67×20⁵²⁸−143)/19<br>(16×20⁴⁴⁹−2809)/19<br>292×20⁴³⁰+7<br>4×20³⁹¹+32091||http://factordb.com/index.php?id=1100000003590539457&open=prime<br>http://factordb.com/index.php?id=1100000002325393915&open=prime<br>http://factordb.com/index.php?id=1100000003590502412&open=prime<br>http://factordb.com/index.php?id=1100000003590502490&open=prime<br>http://factordb.com/index.php?id=1100000001559454258&open=prime<br>http://factordb.com/index.php?id=1100000003590502516&open=prime<br>http://factordb.com/index.php?id=1100000003590502531&open=prime<br>http://factordb.com/index.php?id=1100000000840126753&open=prime<br>http://factordb.com/index.php?id=1100000002633348702&open=prime<br>http://factordb.com/index.php?id=1100000003590502563&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003590539457&base=20<br>http://factordb.com/index.php?showid=1100000002325393915&base=20<br>http://factordb.com/index.php?showid=1100000003590502412&base=20<br>http://factordb.com/index.php?showid=1100000003590502490&base=20<br>http://factordb.com/index.php?showid=1100000001559454258&base=20<br>http://factordb.com/index.php?showid=1100000003590502516&base=20<br>http://factordb.com/index.php?showid=1100000003590502531&base=20<br>http://factordb.com/index.php?showid=1100000000840126753&base=20<br>http://factordb.com/index.php?showid=1100000002633348702&base=20<br>http://factordb.com/index.php?showid=1100000003590502563&base=20<nowiki/>||0||–|| |- ||21||13386~13394||27₁₈₄₄₉₉9D<br>F9₁₇₈₇₇₁D<br>2FC₁₁₂₀₂₂A<br>7₁₀₈₄₅₀ID<br>40₄₇₃₃₃9G<br>B90₄₅₀₁₉E5<br>HD₃₇₄₁₄<br>BD₃₅₀₂₇B<br>990₃₃₂₃₉99H<br>5₃₀₆₀₆FEK||184502<br>178773<br>112025<br>108452<br>47336<br>45023<br>37415<br>35029<br>33244<br>30609||243952<br>236377<br>148121<br>143397<br>62588<br>59531<br>49471<br>46316<br>43956<br>40472||(47×21¹⁸⁴⁵⁰¹+953)/20<br>(309×21¹⁷⁸⁷⁷²+71)/20<br>(288×21¹¹²⁰²³−13)/5<br>(7×21¹⁰⁸⁴⁵²+4733)/20<br>4×21⁴⁷³³⁵+205<br>240×21⁴⁵⁰²¹+299<br>(353×21³⁷⁴¹⁴−13)/20<br>(233×21³⁵⁰²⁸−53)/20<br>198×21³³²⁴²+4175<br>(21³⁰⁶⁰⁹+18455)/4||http://factordb.com/index.php?id=1100000008700600990&open=prime<br>http://factordb.com/index.php?id=1100000008700596669&open=prime<br>http://factordb.com/index.php?id=1100000008700593358&open=prime<br>http://factordb.com/index.php?id=1100000008700586183&open=prime<br>http://factordb.com/index.php?id=1100000000808118331&open=prime<br>http://factordb.com/index.php?id=1100000003996110311&open=prime<br>http://factordb.com/index.php?id=1100000003996110479&open=prime<br>http://factordb.com/index.php?id=1100000003996110718&open=prime<br>http://factordb.com/index.php?id=1100000003996110944&open=prime<br>http://factordb.com/index.php?id=1100000003996111130&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008700600990&base=21<br>http://factordb.com/index.php?showid=1100000008700596669&base=21<br>http://factordb.com/index.php?showid=1100000008700593358&base=21<br>http://factordb.com/index.php?showid=1100000008700586183&base=21<br>http://factordb.com/index.php?showid=1100000000808118331&base=21<br>http://factordb.com/index.php?showid=1100000003996110311&base=21<br>http://factordb.com/index.php?showid=1100000003996110479&base=21<br>http://factordb.com/index.php?showid=1100000003996110718&base=21<br>http://factordb.com/index.php?showid=1100000003996110944&base=21<br>http://factordb.com/index.php?showid=1100000003996111130&base=21<nowiki/>||8||200000|| |- ||22||8003||BK₂₂₀₀₁5<br>7₃₈₁₅2L<br>L₂₃₈₅KE7<br>7₉₅₉K7<br>J0₇₆₇IGGJ<br>K0₇₆₀EC1<br>I₆₂₆AF<br>E60₄₉₆L<br>L₄₈₃G3<br>L0₄₅₄B63||22003<br>3817<br>2388<br>961<br>772<br>764<br>628<br>499<br>485<br>458||29538<br>5124<br>3206<br>1290<br>1037<br>1026<br>843<br>670<br>652<br>615||(251×22²²⁰⁰²−335)/21<br>(22³⁸¹⁷−289)/3<br>22²³⁸⁸−653<br>(22⁹⁶¹+857)/3<br>19×22⁷⁷¹+199779<br>20×22⁷⁶³+7041<br>(6×22⁶²⁸−1259)/7<br>314×22⁴⁹⁷+21<br>22⁴⁸⁵−129<br>21×22⁴⁵⁷+5459||http://factordb.com/index.php?id=1100000003594696838&open=prime<br>http://factordb.com/index.php?id=1100000003591359839&open=prime<br>http://factordb.com/index.php?id=1100000003591360774&open=prime<br>http://factordb.com/index.php?id=1100000003591361817&open=prime<br>http://factordb.com/index.php?id=1100000003591362567&open=prime<br>http://factordb.com/index.php?id=1100000000632724415&open=prime<br>http://factordb.com/index.php?id=1100000000632724334&open=prime<br>http://factordb.com/index.php?id=1100000000632703239&open=prime<br>http://factordb.com/index.php?id=1100000003591364730&open=prime<br>http://factordb.com/index.php?id=1100000003591365331&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003594696838&base=22<br>http://factordb.com/index.php?showid=1100000003591359839&base=22<br>http://factordb.com/index.php?showid=1100000003591360774&base=22<br>http://factordb.com/index.php?showid=1100000003591361817&base=22<br>http://factordb.com/index.php?showid=1100000003591362567&base=22<br>http://factordb.com/index.php?showid=1100000000632724415&base=22<br>http://factordb.com/index.php?showid=1100000000632724334&base=22<br>http://factordb.com/index.php?showid=1100000000632703239&base=22<br>http://factordb.com/index.php?showid=1100000003591364730&base=22<br>http://factordb.com/index.php?showid=1100000003591365331&base=22<nowiki/>||0||–|| |- ||23||65178~65265||B0₉₃₀₄₆FB<br>L₈₆₄₄₄D<br>AJ₈₁₀₆₅4<br>20₇₃₅₆₀98<br>J₆₈₂₁₇G4<br>D70₆₆₇₇₀B<br>5F₆₂₃₄₀6<br>A7M7₆₁₅₃₂<br>B30₆₁₁₃₆5<br>EJ₅₂₁₆₉||93049<br>86445<br>81067<br>73563<br>68219<br>66773<br>62342<br>61535<br>61139<br>52170||126708<br>117715<br>110391<br>100172<br>92896<br>90927<br>84893<br>83794<br>83255<br>71042||11×23⁹³⁰⁴⁸+356<br>(21×23⁸⁶⁴⁴⁵−197)/22<br>(239×23⁸¹⁰⁶⁶−349)/22<br>2×23⁷³⁵⁶²+215<br>(19×23⁶⁸²¹⁹−1867)/22<br>306×23⁶⁶⁷⁷¹+11<br>(125×23⁶²³⁴¹−213)/22<br>(120413×23⁶¹⁵³²−7)/22<br>256×23⁶¹¹³⁷+5<br>(327×23⁵²¹⁶⁹−19)/22||http://factordb.com/index.php?id=1100000004691540361&open=prime<br>http://factordb.com/index.php?id=1100000004691546739&open=prime<br>http://factordb.com/index.php?id=1100000004691548070&open=prime<br>http://factordb.com/index.php?id=1100000004691548569&open=prime<br>http://factordb.com/index.php?id=1100000004691549462&open=prime<br>http://factordb.com/index.php?id=1100000004691549803&open=prime<br>http://factordb.com/index.php?id=1100000004691551005&open=prime<br>http://factordb.com/index.php?id=1100000004691556967&open=prime<br>http://factordb.com/index.php?id=1100000004691557254&open=prime<br>http://factordb.com/index.php?id=1100000004691557548&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004691540361&base=23<br>http://factordb.com/index.php?showid=1100000004691546739&base=23<br>http://factordb.com/index.php?showid=1100000004691548070&base=23<br>http://factordb.com/index.php?showid=1100000004691548569&base=23<br>http://factordb.com/index.php?showid=1100000004691549462&base=23<br>http://factordb.com/index.php?showid=1100000004691549803&base=23<br>http://factordb.com/index.php?showid=1100000004691551005&base=23<br>http://factordb.com/index.php?showid=1100000004691556967&base=23<br>http://factordb.com/index.php?showid=1100000004691557254&base=23<br>http://factordb.com/index.php?showid=1100000004691557548&base=23<nowiki/>||87||100000|| |- ||24||3409||N00N₈₁₂₉LN<br>88N₅₉₅₁<br>A0₂₉₅₁8ID<br>D₂₆₉₈LD<br>N₂₆₄₄LLN<br>BC0₃₃₁B<br>20₃₁₃7<br>C7₂₉₈<br>D0₂₅₉KKD<br>I0₂₄₁I5||8134<br>5953<br>2955<br>2700<br>2647<br>334<br>315<br>299<br>263<br>244||11227<br>8216<br>4079<br>3727<br>3654<br>461<br>434<br>413<br>363<br>337||13249×24⁸¹³¹−49<br>201×24⁵⁹⁵¹−1<br>10×24²⁹⁵⁴+5053<br>(13×24²⁷⁰⁰+4403)/23<br>24²⁶⁴⁷−1201<br>276×24³³²+11<br>2×24³¹⁴+7<br>(283×24²⁹⁸−7)/23<br>13×24²⁶²+12013<br>18×24²⁴³+437||http://factordb.com/index.php?id=1100000003593391606&open=prime<br>http://factordb.com/index.php?id=1100000003593275880&open=prime<br>http://factordb.com/index.php?id=1100000003593269654&open=prime<br>http://factordb.com/index.php?id=1100000003593269876&open=prime<br>http://factordb.com/index.php?id=1100000003593270089&open=prime<br>http://factordb.com/index.php?id=1100000002633359842&open=prime<br>http://factordb.com/index.php?id=1100000002355610241&open=prime<br>http://factordb.com/index.php?id=1100000002326181235&open=prime<br>http://factordb.com/index.php?id=1100000003593270725&open=prime<br>http://factordb.com/index.php?id=1100000002633360037&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000003593391606&base=24<br>http://factordb.com/index.php?showid=1100000003593275880&base=24<br>http://factordb.com/index.php?showid=1100000003593269654&base=24<br>http://factordb.com/index.php?showid=1100000003593269876&base=24<br>http://factordb.com/index.php?showid=1100000003593270089&base=24<br>http://factordb.com/index.php?showid=1100000002633359842&base=24<br>http://factordb.com/index.php?showid=1100000002355610241&base=24<br>http://factordb.com/index.php?showid=1100000002326181235&base=24<br>http://factordb.com/index.php?showid=1100000003593270725&base=24<br>http://factordb.com/index.php?showid=1100000002633360037&base=24<nowiki/>||0||–|| |- ||25||133639~133724||E₉₈₃₉₆FOO<br>1J710₉₆₂₇₂1<br>NB0₈₅₅₉₈5NH<br>D70₈₁₅₈₁JJ7<br>F0₈₀₀₅₄HL<br>J010₇₅₉₄₃E7<br>K₆₇₇₇₁5I<br>LO₆₆₃₇₇KC<br>KJD0₆₃₃₉₉1<br>70₆₀₈₉₂D711||98399<br>96277<br>85603<br>81586<br>80057<br>75948<br>67773<br>66380<br>63403<br>60897||137556<br>134589<br>119668<br>114053<br>111915<br>106171<br>94743<br>92796<br>88634<br>85130||(7×25⁹⁸³⁹⁹+10613)/12<br>27676×25⁹⁶²⁷³+1<br>586×25⁸⁵⁶⁰¹+3717<br>332×25⁸¹⁵⁸⁴+12357<br>15×25⁸⁰⁰⁵⁶+446<br>11876×25⁷⁵⁹⁴⁵+357<br>(5×25⁶⁷⁷⁷³−2267)/6<br>22×25⁶⁶³⁷⁹−113<br>12988×25⁶³⁴⁰⁰+1<br>7×25⁶⁰⁸⁹⁶+207526||http://factordb.com/index.php?id=1100000000808118215&open=prime<br>http://factordb.com/index.php?id=1100000003983674902&open=prime<br>http://factordb.com/index.php?id=1100000004909706420&open=prime<br>http://factordb.com/index.php?id=1100000004909733266&open=prime<br>http://factordb.com/index.php?id=1100000004909750102&open=prime<br>http://factordb.com/index.php?id=1100000004909770736&open=prime<br>http://factordb.com/index.php?id=1100000004586986394&open=prime<br>http://factordb.com/index.php?id=1100000000808118270&open=prime<br>http://factordb.com/index.php?id=1100000004586986664&open=prime<br>http://factordb.com/index.php?id=1100000004586986798&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118215&base=25<br>http://factordb.com/index.php?showid=1100000003983674902&base=25<br>http://factordb.com/index.php?showid=1100000004909706420&base=25<br>http://factordb.com/index.php?showid=1100000004909733266&base=25<br>http://factordb.com/index.php?showid=1100000004909750102&base=25<br>http://factordb.com/index.php?showid=1100000004909770736&base=25<br>http://factordb.com/index.php?showid=1100000004586986394&base=25<br>http://factordb.com/index.php?showid=1100000000808118270&base=25<br>http://factordb.com/index.php?showid=1100000004586986664&base=25<br>http://factordb.com/index.php?showid=1100000004586986798&base=25<nowiki/>||85||100000|| |- ||26||25256~25259||85M₁₉₇₀₆₀B<br>M0₆₁₁₈₆2BB<br>J0₄₄₃₀₃KCB<br>6K₂₃₃₀₀5<br>LD0₂₀₉₇₅7<br>7₂₀₂₇₉OL<br>5₁₉₃₉₁6F<br>9GDK₁₅₉₂₀P<br>M₈₇₇₂P<br>K0₄₃₆₄I5||197063<br>61190<br>44307<br>23302<br>20978<br>20281<br>19393<br>15924<br>8773<br>4367||278839<br>86583<br>62694<br>32972<br>29684<br>28697<br>27440<br>22532<br>12414<br>6180||(5347×26¹⁹⁷⁰⁶¹−297)/25<br>22×26⁶¹¹⁸⁹+1649<br>19×26⁴⁴³⁰⁶+13843<br>(34×26²³³⁰¹−79)/5<br>559×26²⁰⁹⁷⁶+7<br>(7×26²⁰²⁸¹+11393)/25<br>(26¹⁹³⁹³+179)/5<br>(32569×26¹⁵⁹²¹+21)/5<br>(22×26⁸⁷⁷³+53)/25<br>20×26⁴³⁶⁶+473||http://factordb.com/index.php?id=1100000008573990023&open=prime<br>http://factordb.com/index.php?id=1100000003968169875&open=prime<br>http://factordb.com/index.php?id=1100000003968156595&open=prime<br>http://factordb.com/index.php?id=1100000003892628745&open=prime<br>http://factordb.com/index.php?id=1100000003892628658&open=prime<br>http://factordb.com/index.php?id=1100000003892628605&open=prime<br>http://factordb.com/index.php?id=1100000003850151202&open=prime<br>http://factordb.com/index.php?id=1100000003850155316&open=prime<br>http://factordb.com/index.php?id=1100000000758011195&open=prime<br>http://factordb.com/index.php?id=1100000002634136508&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008573990023&base=26<br>http://factordb.com/index.php?showid=1100000003968169875&base=26<br>http://factordb.com/index.php?showid=1100000003968156595&base=26<br>http://factordb.com/index.php?showid=1100000003892628745&base=26<br>http://factordb.com/index.php?showid=1100000003892628658&base=26<br>http://factordb.com/index.php?showid=1100000003892628605&base=26<br>http://factordb.com/index.php?showid=1100000003850151202&base=26<br>http://factordb.com/index.php?showid=1100000003850155316&base=26<br>http://factordb.com/index.php?showid=1100000000758011195&base=26<br>http://factordb.com/index.php?showid=1100000002634136508&base=26<nowiki/>||3||200000|| |- ||27||102852~102896||CA0F₈₈₈₈₃A<br>GNN0₇₈₇₉₅N<br>O44L₆₆₀₁₆7<br>NJ0₆₄₃₆₉H<br>ME₄₉₆₄₀9G<br>PH0₄₇₈₉₀1<br>QF₄₇₁₆₅AF5<br>J0₄₀₇₉₁PD<br>510₃₉₁₆₄I07<br>NGN0₃₆₃₂₉N||88887<br>78799<br>66020<br>64372<br>49643<br>47893<br>47169<br>40794<br>39169<br>36333||127230<br>112790<br>94499<br>92140<br>71058<br>68553<br>67516<br>58391<br>56065<br>52006||(234483×27⁸⁸⁸⁸⁴−145)/26<br>12308×27⁷⁸⁷⁹⁶+23<br>(457829×27⁶⁶⁰¹⁷−385)/26<br>640×27⁶⁴³⁷⁰+17<br>(293×27⁴⁹⁶⁴²−1736)/13<br>692×27⁴⁷⁸⁹¹+1<br>(691×27⁴⁷¹⁶⁸−95045)/26<br>19×27⁴⁰⁷⁹³+688<br>136×27³⁹¹⁶⁷+13129<br>17222×27³⁶³³⁰+23||http://factordb.com/index.php?id=1100000000808118233&open=prime<br>http://factordb.com/index.php?id=1100000004681348398&open=prime<br>http://factordb.com/index.php?id=1100000004374140861&open=prime<br>http://factordb.com/index.php?id=1100000004374138999&open=prime<br>http://factordb.com/index.php?id=1100000000819229859&open=prime<br>http://factordb.com/index.php?id=1100000004102754118&open=prime<br>http://factordb.com/index.php?id=1100000004102755880&open=prime<br>http://factordb.com/index.php?id=1100000004102758254&open=prime<br>http://factordb.com/index.php?id=1100000004102875088&open=prime<br>http://factordb.com/index.php?id=1100000004103372866&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118233&base=27<br>http://factordb.com/index.php?showid=1100000004681348398&base=27<br>http://factordb.com/index.php?showid=1100000004374140861&base=27<br>http://factordb.com/index.php?showid=1100000004374138999&base=27<br>http://factordb.com/index.php?showid=1100000000819229859&base=27<br>http://factordb.com/index.php?showid=1100000004102754118&base=27<br>http://factordb.com/index.php?showid=1100000004102755880&base=27<br>http://factordb.com/index.php?showid=1100000004102758254&base=27<br>http://factordb.com/index.php?showid=1100000004102875088&base=27<br>http://factordb.com/index.php?showid=1100000004103372866&base=27<nowiki/>||44||100000|| |- ||28||25528~25529||O4O₉₄₅₃₅9<br>5OA₃₁₂₃₈F<br>N6₂₄₀₅₁LR<br>D0₅₂₆₇77D<br>QO₄₂₃₉69<br>5₃₇₄₆8P<br>G0₁₈₉₉AN<br>A₁₄₂₃6F<br>5I₁₃₇₀F<br>5₁₃₃₂P8P||94538<br>31241<br>24054<br>5271<br>4242<br>3748<br>1902<br>1425<br>1372<br>1335||136812<br>45210<br>34810<br>7628<br>6139<br>5424<br>2753<br>2062<br>1985<br>1932||(6092×28⁹⁴⁵³⁶−143)/9<br>(4438×28³¹²³⁹+125)/27<br>(209×28²⁴⁰⁵³+3967)/9<br>13×28⁵²⁷⁰+5697<br>(242×28⁴²⁴¹−4679)/9<br>(5×28³⁷⁴⁸+2803)/27<br>16×28¹⁹⁰¹+303<br>(10×28¹⁴²⁵−2899)/27<br>(17×28¹³⁷¹−11)/3<br>(5×28¹³³⁵+426163)/27||http://factordb.com/index.php?id=1100000000808118231&open=prime<br>http://factordb.com/index.php?id=1100000003880455200&open=prime<br>http://factordb.com/index.php?id=1100000003879667576&open=prime<br>http://factordb.com/index.php?id=1100000003850151420&open=prime<br>http://factordb.com/index.php?id=1100000000840839934&open=prime<br>http://factordb.com/index.php?id=1100000003850161974&open=prime<br>http://factordb.com/index.php?id=1100000003850161973&open=prime<br>http://factordb.com/index.php?id=1100000000840839947&open=prime<br>http://factordb.com/index.php?id=1100000003850161972&open=prime<br>http://factordb.com/index.php?id=1100000003850161965&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000000808118231&base=28<br>http://factordb.com/index.php?showid=1100000003880455200&base=28<br>http://factordb.com/index.php?showid=1100000003879667576&base=28<br>http://factordb.com/index.php?showid=1100000003850151420&base=28<br>http://factordb.com/index.php?showid=1100000000840839934&base=28<br>http://factordb.com/index.php?showid=1100000003850161974&base=28<br>http://factordb.com/index.php?showid=1100000003850161973&base=28<br>http://factordb.com/index.php?showid=1100000000840839947&base=28<br>http://factordb.com/index.php?showid=1100000003850161972&base=28<br>http://factordb.com/index.php?showid=1100000003850161965&base=28<nowiki/>||1||900000|| |- ||29||355242~355367||830₉₉₃₇₇4<br>GP5J₉₄₉₃₅<br>P05J₉₀₂₈₉<br>BBD0₈₈₈₈₈PB<br>8B₈₅₃₃₃G<br>L0₈₁₅₇₁5955<br>E0₇₇₃₇₂L7B<br>LPC₇₅₁₅₁9<br>JR0₇₄₆₂₂7<br>B₇₄₅₀₁0RP||99380<br>94938<br>90292<br>88893<br>85335<br>81576<br>77376<br>75154<br>74625<br>74504||145333<br>138837<br>132043<br>129997<br>124794<br>119297<br>113155<br>109905<br>109132<br>108955||235×29⁹⁹³⁷⁸+4<br>(397227×29⁹⁴⁹³⁵−19)/28<br>(588859×29⁹⁰²⁸⁹−19)/28<br>9583×29⁸⁸⁸⁹⁰+736<br>(235×29⁸⁵³³⁴+129)/14<br>21×29⁸¹⁵⁷⁵+129664<br>14×29⁷⁷³⁷⁵+17875<br>(4441×29⁷⁵¹⁵²−24)/7<br>578×29⁷⁴⁶²³+7<br>(11×29⁷⁴⁵⁰⁴−245655)/28||http://factordb.com/index.php?id=1100000008253882372&open=prime<br>http://factordb.com/index.php?id=1100000008253893542&open=prime<br>http://factordb.com/index.php?id=1100000008253899083&open=prime<br>http://factordb.com/index.php?id=1100000008253909183&open=prime<br>http://factordb.com/index.php?id=1100000008253921388&open=prime<br>http://factordb.com/index.php?id=1100000008253925955&open=prime<br>http://factordb.com/index.php?id=1100000008253931446&open=prime<br>http://factordb.com/index.php?id=1100000000808118236&open=prime<br>http://factordb.com/index.php?id=1100000008253934219&open=prime<br>http://factordb.com/index.php?id=1100000008253936120&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008253882372&base=29<br>http://factordb.com/index.php?showid=1100000008253893542&base=29<br>http://factordb.com/index.php?showid=1100000008253899083&base=29<br>http://factordb.com/index.php?showid=1100000008253909183&base=29<br>http://factordb.com/index.php?showid=1100000008253921388&base=29<br>http://factordb.com/index.php?showid=1100000008253925955&base=29<br>http://factordb.com/index.php?showid=1100000008253931446&base=29<br>http://factordb.com/index.php?showid=1100000000808118236&base=29<br>http://factordb.com/index.php?showid=1100000008253934219&base=29<br>http://factordb.com/index.php?showid=1100000008253936120&base=29<nowiki/>||125||100000|| |- ||30||2619||OT₃₄₂₀₅<br>I0₂₄₆₀₈D<br>5₄₈₈₂J<br>C0₁₀₂₂1<br>M0₅₄₇SS7<br>M₂₄₁QB<br>AN₂₀₆<br>50₁₆₄B<br>J₁₅₃QJ<br>J₉₄QQJ||34206<br>24610<br>4883<br>1024<br>551<br>243<br>207<br>166<br>155<br>97||50527<br>36352<br>7213<br>1513<br>814<br>359<br>306<br>245<br>229<br>144||25×30³⁴²⁰⁵−1<br>18×30²⁴⁶⁰⁹+13<br>(5×30⁴⁸⁸³+401)/29<br>12×30¹⁰²³+1<br>22×30⁵⁵⁰+26047<br>(22×30²⁴³+3139)/29<br>(313×30²⁰⁶−23)/29<br>5×30¹⁶⁵+11<br>(19×30¹⁵⁵+6071)/29<br>(19×30⁹⁷+188771)/29||http://factordb.com/index.php?id=1100000000800812865&open=prime<br>http://factordb.com/index.php?id=1100000003593967511&open=prime<br>http://factordb.com/index.php?id=1100000002327649423&open=prime<br>http://factordb.com/index.php?id=1100000000785448736&open=prime<br>http://factordb.com/index.php?id=1100000003593407988&open=prime<br>http://factordb.com/index.php?id=1100000003593408295&open=prime<br>http://factordb.com/index.php?id=1100000002327651073&open=prime<br>http://factordb.com/index.php?id=1100000002356282476&open=ecm<br>http://factordb.com/index.php?id=1100000003593409109&open=ecm<br>http://factordb.com/index.php?id=1100000003593409165&open=ecm<nowiki/>||http://factordb.com/index.php?showid=1100000000800812865&base=30<br>http://factordb.com/index.php?showid=1100000003593967511&base=30<br>http://factordb.com/index.php?showid=1100000002327649423&base=30<br>http://factordb.com/index.php?showid=1100000000785448736&base=30<br>http://factordb.com/index.php?showid=1100000003593407988&base=30<br>http://factordb.com/index.php?showid=1100000003593408295&base=30<br>http://factordb.com/index.php?showid=1100000002327651073&base=30<br>http://factordb.com/index.php?showid=1100000002356282476&base=30<br>http://factordb.com/index.php?showid=1100000003593409109&base=30<br>http://factordb.com/index.php?showid=1100000003593409165&base=30<nowiki/>||0||–|| |- ||31||569323~569400||2IIF₉₁₈₀₅<br>B0₈₈₃₀₉APO9<br>J0T₇₇₅₁₆<br>J090₇₇₁₂₈NNN<br>D₆₉₈₆₁QO<br>9MH0₆₈₆₃₇D<br>J₆₇₁₆₂D<br>N0₆₆₉₇₁32P<br>DDDQ0₆₄₀₈₈TD<br>U₆₃₈₆₁CM3||91808<br>88314<br>77518<br>77134<br>69863<br>68641<br>67163<br>66975<br>64094<br>63864||136918<br>131708<br>115608<br>115035<br>104191<br>102369<br>100165<br>99884<br>95587<br>95245||(4997×31⁹¹⁸⁰⁵−1)/2<br>11×31⁸⁸³¹³+322688<br>(17699×31⁷⁷⁵¹⁶−29)/30<br>18268×31⁷⁷¹³¹+22839<br>(13×31⁶⁹⁸⁶³+12407)/30<br>9348×31⁶⁸⁶³⁸+13<br>(19×31⁶⁷¹⁶³−199)/30<br>23×31⁶⁶⁹⁷⁴+2970<br>400205×31⁶⁴⁰⁹⁰+912<br>31⁶³⁸⁶⁴−17574||http://factordb.com/index.php?id=1100000007050395732&open=prime<br>http://factordb.com/index.php?id=1100000007050397309&open=prime<br>http://factordb.com/index.php?id=1100000007050398940&open=prime<br>http://factordb.com/index.php?id=1100000007050400178&open=prime<br>http://factordb.com/index.php?id=1100000006965878559&open=prime<br>http://factordb.com/index.php?id=1100000006965875678&open=prime<br>http://factordb.com/index.php?id=1100000006965873668&open=prime<br>http://factordb.com/index.php?id=1100000006965870538&open=prime<br>http://factordb.com/index.php?id=1100000006965868103&open=prime<br>http://factordb.com/index.php?id=1100000006965865343&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000007050395732&base=31<br>http://factordb.com/index.php?showid=1100000007050397309&base=31<br>http://factordb.com/index.php?showid=1100000007050398940&base=31<br>http://factordb.com/index.php?showid=1100000007050400178&base=31<br>http://factordb.com/index.php?showid=1100000006965878559&base=31<br>http://factordb.com/index.php?showid=1100000006965875678&base=31<br>http://factordb.com/index.php?showid=1100000006965873668&base=31<br>http://factordb.com/index.php?showid=1100000006965870538&base=31<br>http://factordb.com/index.php?showid=1100000006965868103&base=31<br>http://factordb.com/index.php?showid=1100000006965865343&base=31<nowiki/>||77||100000|| |- ||32||168882~169002||V₉₉₅₈₃63<br>6₈₉₀₇₄AF<br>8₇₇₇₀₀H<br>Q₇₇₄₀₁EQQQ3<br>8₇₇₂₄₉3<br>JM₇₆₀₂₈L<br>E₇₂₉₁₉IL<br>B0₆₇₆₈₀CB<br>GK₆₆₀₇₆F<br>KN₆₅₀₂₂||99585<br>89076<br>77701<br>77406<br>77250<br>76030<br>72921<br>67683<br>66078<br>65023||149891<br>134073<br>116952<br>116508<br>116273<br>114437<br>109757<br>101873<br>99458<br>97870||32⁹⁹⁵⁸⁵−829<br>(6×32⁸⁹⁰⁷⁶+4241)/31<br>(8×32⁷⁷⁷⁰¹+271)/31<br>(26×32⁷⁷⁴⁰⁶−390071011)/31<br>(8×32⁷⁷²⁵⁰−163)/31<br>(611×32⁷⁶⁰²⁹−53)/31<br>(14×32⁷²⁹²¹+4171)/31<br>11×32⁶⁷⁶⁸²+395<br>(516×32⁶⁶⁰⁷⁷−175)/31<br>(643×32⁶⁵⁰²²−23)/31||http://factordb.com/index.php?id=1100000005514892191&open=prime<br>http://factordb.com/index.php?id=1100000005514897129&open=prime<br>http://factordb.com/index.php?id=1100000005514901700&open=prime<br>http://factordb.com/index.php?id=1100000005514915338&open=prime<br>http://factordb.com/index.php?id=1100000005514918574&open=prime<br>http://factordb.com/index.php?id=1100000005514922523&open=prime<br>http://factordb.com/index.php?id=1100000004591654373&open=prime<br>http://factordb.com/index.php?id=1100000004591654467&open=prime<br>http://factordb.com/index.php?id=1100000004591654632&open=prime<br>http://factordb.com/index.php?id=1100000004591654952&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005514892191&base=32<br>http://factordb.com/index.php?showid=1100000005514897129&base=32<br>http://factordb.com/index.php?showid=1100000005514901700&base=32<br>http://factordb.com/index.php?showid=1100000005514915338&base=32<br>http://factordb.com/index.php?showid=1100000005514918574&base=32<br>http://factordb.com/index.php?showid=1100000005514922523&base=32<br>http://factordb.com/index.php?showid=1100000004591654373&base=32<br>http://factordb.com/index.php?showid=1100000004591654467&base=32<br>http://factordb.com/index.php?showid=1100000004591654632&base=32<br>http://factordb.com/index.php?showid=1100000004591654952&base=32<nowiki/>||120||100000|| |- ||33||280012~280093||DP₉₅₀₉₃M5<br>HJ0₉₄₂₉₅J<br>90₉₃₅₉₇Q<br>9F0₉₃₁₅₇N<br>7₈₉₄₄₉333H<br>K3₈₀₇₅₁6K<br>D₈₀₁₀₇9UD<br>VFU₇₂₂₀₄FK<br>J₆₈₇₁₅2BJ<br>DF0₆₈₃₆₇J||95096<br>94298<br>93599<br>93160<br>89453<br>80754<br>80110<br>72208<br>68718<br>68370||144405<br>143193<br>142131<br>141465<br>135835<br>122626<br>121648<br>109649<br>104350<br>103821||(441×33⁹⁵⁰⁹⁵−3833)/32<br>580×33⁹⁴²⁹⁶+19<br>9×33⁹³⁵⁹⁸+26<br>312×33⁹³¹⁵⁸+23<br>(7×33⁸⁹⁴⁵³−4743239)/32<br>(643×33⁸⁰⁷⁵³+3709)/32<br>(13×33⁸⁰¹¹⁰−121453)/32<br>(16623×33⁷²²⁰⁶−8095)/16<br>(19×33⁶⁸⁷¹⁸−600883)/32<br>444×33⁶⁸³⁶⁸+19||http://factordb.com/index.php?id=1100000005652348775&open=prime<br>http://factordb.com/index.php?id=1100000005652362811&open=prime<br>http://factordb.com/index.php?id=1100000005652375073&open=prime<br>http://factordb.com/index.php?id=1100000005652389776&open=prime<br>http://factordb.com/index.php?id=1100000005652430746&open=prime<br>http://factordb.com/index.php?id=1100000005652446200&open=prime<br>http://factordb.com/index.php?id=1100000005652461592&open=prime<br>http://factordb.com/index.php?id=1100000004614764298&open=prime<br>http://factordb.com/index.php?id=1100000004614770536&open=prime<br>http://factordb.com/index.php?id=1100000004614784274&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000005652348775&base=33<br>http://factordb.com/index.php?showid=1100000005652362811&base=33<br>http://factordb.com/index.php?showid=1100000005652375073&base=33<br>http://factordb.com/index.php?showid=1100000005652389776&base=33<br>http://factordb.com/index.php?showid=1100000005652430746&base=33<br>http://factordb.com/index.php?showid=1100000005652446200&base=33<br>http://factordb.com/index.php?showid=1100000005652461592&base=33<br>http://factordb.com/index.php?showid=1100000004614764298&base=33<br>http://factordb.com/index.php?showid=1100000004614770536&base=33<br>http://factordb.com/index.php?showid=1100000004614784274&base=33<nowiki/>||81||100000|| |- ||34||184785~184832||GFGC₉₉₉₉₆5<br>90₉₇₉₅₀FJ<br>NM0₈₅₂₁₈KX<br>F₈₃₁₈₉H2HP<br>P₇₉₄₄₁444P<br>6₇₇₀₂₇8X<br>XQIQ₇₂₂₄₁D<br>T₆₆₅₃₀IF<br>4₆₆₁₅₂B<br>2EEC₆₆₀₃₉7||100000<br>97953<br>85222<br>83193<br>79445<br>77029<br>72245<br>66532<br>66153<br>66043||153148<br>150013<br>130516<br>127408<br>121669<br>117968<br>110642<br>101893<br>101312<br>101143||(209246×34⁹⁹⁹⁹⁷−81)/11<br>9×34⁹⁷⁹⁵²+529<br>804×34⁸⁵²²⁰+713<br>(5×34⁸³¹⁹³+700233)/11<br>(25×34⁷⁹⁴⁴⁵−28062367)/33<br>(2×34⁷⁷⁰²⁹+1043)/11<br>(1288676×34⁷²²⁴²−455)/33<br>(29×34⁶⁶⁵³²−12833)/33<br>(4×34⁶⁶¹⁵³+227)/33<br>(30826×34⁶⁶⁰⁴⁰−59)/11||http://factordb.com/index.php?id=1100000004702891268&open=prime<br>http://factordb.com/index.php?id=1100000004702894713&open=prime<br>http://factordb.com/index.php?id=1100000004702900996&open=prime<br>http://factordb.com/index.php?id=1100000004702910376&open=prime<br>http://factordb.com/index.php?id=1100000004702913746&open=prime<br>http://factordb.com/index.php?id=1100000004702918600&open=prime<br>http://factordb.com/index.php?id=1100000004399656529&open=prime<br>http://factordb.com/index.php?id=1100000004399657696&open=prime<br>http://factordb.com/index.php?id=1100000004399658651&open=prime<br>http://factordb.com/index.php?id=1100000004399659716&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000004702891268&base=34<br>http://factordb.com/index.php?showid=1100000004702894713&base=34<br>http://factordb.com/index.php?showid=1100000004702900996&base=34<br>http://factordb.com/index.php?showid=1100000004702910376&base=34<br>http://factordb.com/index.php?showid=1100000004702913746&base=34<br>http://factordb.com/index.php?showid=1100000004702918600&base=34<br>http://factordb.com/index.php?showid=1100000004399656529&base=34<br>http://factordb.com/index.php?showid=1100000004399657696&base=34<br>http://factordb.com/index.php?showid=1100000004399658651&base=34<br>http://factordb.com/index.php?showid=1100000004399659716&base=34<nowiki/>||47||100000|| |- ||35||720002~720062||N0N₉₉₉₇₁9<br>V0₈₃₆₆₉E73<br>N₈₁₅₆₃K7N<br>BJ0₈₁₂₇₉N<br>J0₈₀₀₆₂FUH<br>43V₇₉₇₅₄<br>9₇₆₆₀₀K3<br>LB₇₁₃₆₆PB<br>Q₆₄₁₅₀H<br>50₆₃₃₉₇5R||99974<br>83673<br>81566<br>81282<br>80066<br>79756<br>76602<br>71369<br>64151<br>63400||154367<br>129197<br>125944<br>125505<br>123628<br>123148<br>118279<br>110199<br>99054<br>97894||(27393×35⁹⁹⁹⁷²−499)/34<br>31×35⁸³⁶⁷²+17398<br>(23×35⁸¹⁵⁶⁶−144013)/34<br>404×35⁸¹²⁸⁰+23<br>19×35⁸⁰⁰⁶⁵+19442<br>(4893×35⁷⁹⁷⁵⁴−31)/34<br>(9×35⁷⁶⁶⁰²+12877)/34<br>(725×35⁷¹³⁶⁸+16649)/34<br>(13×35⁶⁴¹⁵¹−166)/17<br>5×35⁶³³⁹⁹+202||http://factordb.com/index.php?id=1100000008248342445&open=prime<br>http://factordb.com/index.php?id=1100000008248353306&open=prime<br>http://factordb.com/index.php?id=1100000008248375642&open=prime<br>http://factordb.com/index.php?id=1100000008248397018&open=prime<br>http://factordb.com/index.php?id=1100000008248412468&open=prime<br>http://factordb.com/index.php?id=1100000008248418540&open=prime<br>http://factordb.com/index.php?id=1100000008248423670&open=prime<br>http://factordb.com/index.php?id=1100000008192119974&open=prime<br>http://factordb.com/index.php?id=1100000008192126630&open=prime<br>http://factordb.com/index.php?id=1100000008192129294&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000008248342445&base=35<br>http://factordb.com/index.php?showid=1100000008248353306&base=35<br>http://factordb.com/index.php?showid=1100000008248375642&base=35<br>http://factordb.com/index.php?showid=1100000008248397018&base=35<br>http://factordb.com/index.php?showid=1100000008248412468&base=35<br>http://factordb.com/index.php?showid=1100000008248418540&base=35<br>http://factordb.com/index.php?showid=1100000008248423670&base=35<br>http://factordb.com/index.php?showid=1100000008192119974&base=35<br>http://factordb.com/index.php?showid=1100000008192126630&base=35<br>http://factordb.com/index.php?showid=1100000008192129294&base=35<nowiki/>||60||100000|| |- ||36||35286~35290||P₈₁₉₉₃SZ<br>S0₇₅₀₀₇8H<br>7K₂₆₅₆₇Z<br>J₁₀₁₁₇LJ<br>VL0₇₂₅₈J<br>EO0₆₁₇₇V<br>FZ₅₇₇₇3P<br>T09₄₆₁₈1<br>RY₄₅₆₂H<br>OZ₃₉₃₂AZ||81995<br>75010<br>26569<br>10119<br>7261<br>6180<br>5780<br>4621<br>4564<br>3935||127609<br>116739<br>41349<br>15748<br>11301<br>9618<br>8996<br>7192<br>7103<br>6124||(5×36⁸¹⁹⁹⁵+821)/7<br>28×36⁷⁵⁰⁰⁹+305<br>(53×36²⁶⁵⁶⁸+101)/7<br>(19×36¹⁰¹¹⁹+2501)/35<br>1137×36⁷²⁵⁹+19<br>528×36⁶¹⁷⁸+31<br>16×36⁵⁷⁷⁹−1163<br>(36549×36⁴⁶¹⁹−289)/35<br>(979×36⁴⁵⁶³−629)/35<br>25×36³⁹³⁴−901||http://factordb.com/index.php?id=1100000002394962083&open=prime<br>http://factordb.com/index.php?id=1100000004020085177&open=prime<br>http://factordb.com/index.php?id=1100000003896952461&open=prime<br>http://factordb.com/index.php?id=1100000003807362491&open=prime<br>http://factordb.com/index.php?id=1100000003807362489&open=prime<br>http://factordb.com/index.php?id=1100000003807362488&open=prime<br>http://factordb.com/index.php?id=1100000003807362487&open=prime<br>http://factordb.com/index.php?id=1100000003807362486&open=prime<br>http://factordb.com/index.php?id=1100000003807362485&open=prime<br>http://factordb.com/index.php?id=1100000000840634476&open=prime<nowiki/>||http://factordb.com/index.php?showid=1100000002394962083&base=36<br>http://factordb.com/index.php?showid=1100000004020085177&base=36<br>http://factordb.com/index.php?showid=1100000003896952461&base=36<br>http://factordb.com/index.php?showid=1100000003807362491&base=36<br>http://factordb.com/index.php?showid=1100000003807362489&base=36<br>http://factordb.com/index.php?showid=1100000003807362488&base=36<br>http://factordb.com/index.php?showid=1100000003807362487&base=36<br>http://factordb.com/index.php?showid=1100000003807362486&base=36<br>http://factordb.com/index.php?showid=1100000003807362485&base=36<br>http://factordb.com/index.php?showid=1100000000840634476&base=36<nowiki/>||4||200000|| |} == The fully proof of Athena problem in decimal (base ''b'' = 10) == '''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''. Assume ''p'' is a prime > 10, and the last digit of ''p'' must lie in {1,3,7,9}. Case 1: ''p'' ends with 1. In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9. Case 1.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''. If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''. Case 1.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8. If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}. If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''. If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite. Case 1.3: ''p'' begins with 8. In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5. If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}. If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}. If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite. If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element. Case 1.4: ''p'' begins with 9. In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8. If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8. If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}. If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction). If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801. Case 2: ''p'' ends with 3. In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 3: ''p'' ends with 7. In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8. Case 3.1: ''p'' begins with 2. In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8. If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}. If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime. If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40_''n''1 = 280_''n''7. Case 3.2: ''p'' begins with 5. In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2. If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's. If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime. If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty. If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''. If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''. Case 3.3: ''p'' begins with 7. In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime. Case 3.4: ''p'' begins with 8. In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime. Case 4: ''p'' ends with 9. In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9. If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's. If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4. Case 4.1: ''p'' begins with 3. In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''. Case 4.2: ''p'' begins with 4. In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6_''n''7 = 46_''n''9. Case 4.3: ''p'' begins with 6. In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty. If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6. If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}. If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes. Case 4.4: ''p'' begins with 9. In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669. [[Category:Number theory]] hxefvb1k07di258laiqhjqy1g97qzi8 0 329922 2819649 2818373 2026-07-28T13:47:39Z Mrchristian 281704 2819649 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 <nowiki>https://kisski.gwdg.de/leistungen/2-02-llm-service/</nowiki> | https://chat-ai.academiccloud.de/chat lm25yi5bkgm8pk9b8ozave9pliqnz1i 2819650 2819649 2026-07-28T13:50:57Z Mrchristian 281704 2819650 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat | https://m365.cloud.microsoft/chat dpen9pvglu4ww25ya1ohishiey4rnyd 2819661 2819650 2026-07-28T15:00:47Z Mrchristian 281704 2819661 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat | https://m365.cloud.microsoft/chat == 7. Completion of project section of Linked Open Exhibition == The three sections: ### Wikidata Exhibition entries - Task issue on GitHub https://github.com/NFDI4Culture/linked-open-exhibition/issues/6 ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining rnpnyosw2qiydl9bgzcsdb70zhxnh5y 2819662 2819661 2026-07-28T15:02:10Z Mrchristian 281704 2819662 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat | https://m365.cloud.microsoft/chat == 7. Completion of project section of Linked Open Exhibition == The three sections: # Wikidata Exhibition entries - Task issue on GitHub https://github.com/NFDI4Culture/linked-open-exhibition/issues/6 # DNB (Library metadata) entries sorting # Exhibition catalogue scan - Text and Data Mining mzafk69u2bcsei7tqhnnadio6k3ee3v 2819668 2819662 2026-07-28T15:08:33Z Mrchristian 281704 2819668 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat | https://m365.cloud.microsoft/chat == 7. Completion of project section of Linked Open Exhibition == The three sections: 1. Wikidata Exhibition entries; 2. DNB (Library metadata), and; 3. HsH Exhibition catalogue scan. Crowdsourcing tasks. For the three prototype team we are working with the idea of crowdsourcing. This is where contributors carry out small tasks and by distributing the tasks large scale jobs are broken down and can be more easily completed. All team members need to carry out the same crowdsourcing task, linked here as GitHub issues. These crowdsoucing tasks will contribute to data quality via review and allowing us to see if there are connections beween our data source: Wikidata, the National Library, and the local library. # Wikidata Exhibition entries - Task issue on GitHub https://github.com/NFDI4Culture/linked-open-exhibition/issues/6 # DNB (Library metadata) entries sorting # Exhibition catalogue scan - Text and Data Mining 2fx7kwpmzw6kp2vies6i9g4p20j613y 2819681 2819668 2026-07-28T15:31:30Z Mrchristian 281704 2819681 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat | https://m365.cloud.microsoft/chat == 7. Completion of project section of Linked Open Exhibition == The three sections: 1. Wikidata Exhibition entries; 2. DNB (Library metadata), and; 3. HsH Exhibition catalogue scan. Crowdsourcing tasks. For the three prototype team we are working with the idea of crowdsourcing. This is where contributors carry out small tasks and by distributing the tasks large scale jobs are broken down and can be more easily completed. All team members need to carry out the same crowdsourcing task, linked here as GitHub issues. These crowdsoucing tasks will contribute to data quality via review and allowing us to see if there are connections beween our data source: Wikidata, the National Library, and the local library. # Wikidata Exhibition entries - Task issue on GitHub: https://github.com/NFDI4Culture/linked-open-exhibition/issues/6 # DNB (Library metadata) entries sorting - Task issue on GitHub: https://github.com/NFDI4Culture/linked-open-exhibition/issues/7 # Exhibition catalogue scan - Text and Data Mining - Task issue on GitHub: b1ozynf2xc222reuvwnmdphbuxj7qje 2819686 2819681 2026-07-28T15:44:47Z Mrchristian 281704 2819686 wikitext text/x-wiki Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/ Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]] DE version - see language switcher - top right. Tasks: # Complete the Wikidata entry for a Sprengel Museum exhibition # Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype # Adding Data Model mapping to standards to forked repository # Adding SPARQL Query network diagram to forked repository # Adding ORCID ID to forked repository # AI LLMs: ## Agentic coding: VSCode Copilot exercise ## Document AI LLM use with list of use, pro and cons, and attribution # Completion of project section of Linked Open Exhibitions ## The three sections: ### Wikidata Exhibition entries ### DNB (Library metadata) entries sorting ### Exhibition catalogue scan - Text and Data Mining --- == 1: Complete the Wikidata entry for a Sprengel Museum exhibition == [[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]] [[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]] [[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers: * How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc. For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982 and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition. * View the results collected for the exhibition record: ** In the Query Service results link that shows all of your entries - example Figure 1, ** as a timeline: https://w.wiki/J8NJ - example Figure 2. and ** as a graph: https://w.wiki/J8aS - example Figure 3. * Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics: # Wikidata good practice # Consulting schemas # Importance of review # Using GitHub Issues # Comparing available data – before and after ==== Step-by-step guide: ==== ==== A. Creating the exhibition entry in Wikidata. ==== # Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982 # Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g., #* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records. #** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true #** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026) #** https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised (offline July 2026) # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] # References can be added: Source URLs, date accessed ===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' ===== {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} Task #1 complete! --- == 2. Completion of the GitHub task of forking repository and publishing Wikidata entry == [[File:Wikidata 2026 06 02.jpg|left|thumb]] Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- == 3. Adding Data Model mapping to standards to forked repository == Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/ Choose data models that relate to your Wikidata entry. Data models are: * Artist Data Model * Exhibition Data Model * DNB Catalogue Data Model * Item in Exhibition Data Model Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so: website:   <code>title: "BIM Prototype 02"</code> <code>  navbar:</code> <code>    left:</code> <code>          - href: artist-datamodel.qmd</code> <code>            text: Artist Data Model</code> <code>          - href: exhibition-datamodel.qmd</code> <code>            text: Exhibition Data Model</code> <code>          - href: dnb-catalogue-datamodel.qmd</code> <code>            text: DNB Catalogue Data Model</code> <code>          - href: item-in-exhibition-datamodel.qmd</code> <code>            text: Item in Exhibition Data Model</code> == 4. Adding SPARQL Query network diagram to forked repository == '''Visualizing the Wikidata Item as a Graph''' https://github.com/mrchristian/prototype The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties. In your Quarto project the Jupyter Lab Notebook will render the graph automatically<blockquote>wikidata-item.ipynb</blockquote> # In cell 2 input your Wikidata QID, e.g., item_id = "Q138572982" # Click Run All at the top of the Jupyter Lab Notebook. The graph will then render. # Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub. [[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]] == 5. Adding ORCID ID to forked repository == '''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: <code>XXXX-XXXX-XXXX-XXXX</code> See full details here: https://nfdi4culture.github.io/linked-open-exhibition/ ==== How to Get an ORCID ==== # '''Visit''': orcid.org # '''Click''': “Sign in” → “Register for an ORCID iD” # '''Provide''': #* Given name and family name #* Email address #* Password #* Affiliation (optional but recommended) # '''Verify''': Confirm your email address # '''Complete''': Your 16-digit ORCID will be generated immediately ==== Add to Quarto ==== _quarto.yml <code>project''':'''</code> <code>type''':''' website</code> <code>title''':''' "My Project"</code> <code>metadata''':'''</code> <code>author''':'''</code> <code>'''-''' name''':''' Jane Researcher</code> <code>- orcid''':''' 0000-0002-1234-5678</code> ==== Add to CFF Citation File Format ==== This will make your repository citable on GitHub. Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID. == 6. AI LLM: Agentic assistent/coding == For the project Copilot is used in VSCode for limited agentic coding. A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used. Once logged into VSCode, see the menu item: View > Chat to access the AI on the right. Use Agent mode. ==== Exercises: ==== # Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID <code>XXXX-XXXX-XXXX-XXXX</code> # Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project. # Ask the agent to render and push your Auarto project to Git. ==== Request an account with KISSKI this can be used later for code and questions. ==== „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat | https://m365.cloud.microsoft/chat == 7. Completion of project section of Linked Open Exhibition == The three sections: 1. Wikidata Exhibition entries; 2. DNB (Library metadata), and; 3. HsH Exhibition catalogue scan. Crowdsourcing tasks. For the three prototype team we are working with the idea of crowdsourcing. This is where contributors carry out small tasks and by distributing the tasks large scale jobs are broken down and can be more easily completed. All team members need to carry out the same crowdsourcing task, linked here as GitHub issues. These crowdsoucing tasks will contribute to data quality via review and allowing us to see if there are connections beween our data source: Wikidata, the National Library, and the local library. # Wikidata Exhibition entries - Task issue on GitHub: https://github.com/NFDI4Culture/linked-open-exhibition/issues/6 # DNB (Library metadata) entries sorting - Task issue on GitHub: https://github.com/NFDI4Culture/linked-open-exhibition/issues/7 # Exhibition catalogue scan - Text and Data Mining - Task issue on GitHub: https://github.com/NFDI4Culture/linked-open-exhibition/issues/8 gmxxo232rc0tymnbpkkwfk11tplq474 0 330494 2819642 2819410 2026-07-28T12:09:16Z NDM2024 2984088 2819642 wikitext text/x-wiki '''<big>The John Snow Prediabetes Institute.</big>''' https://w.wiki/Skm7 The John Snow prediabetes Institute is an international research network focused on prediabetes (prevention) remission, early risk identification, and metabolic health education. <big>Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. Early diagnosis and education of prediabetes is a cost-effective preventive strategy that can improve long-term health outcomes. A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained. For early identification of the risks we propose to register weight and height (BMI), the fasting blood sugar (glucometer), blood pressure, age, gender in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools followed by giving educational materials.The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer and self-evaluation of diet and physical activity. Educational materials is available from the international diabetes organisations e.g from the ADA: <ref>https://professional.diabetes.org/diabetes-support-resources</ref>[[File:Lifestyle Medicine Pillars.png|300px|right|<big>The focus of Lifestyle Medicine is on these 6 pillars.</big>]] [[File:Cholera in London 1866.gif|thumb|250px|<big>Map of a later cholera outbreak in London, in 1866</big>]] [[File:Choleramaplondon1866.png|thumb|right|250px|<big>Legend for the map above</big>]]<big>1. '<nowiki/>'''Prevalence studies''''</big> <big>1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref></big> <big>1.2 The Maritime Officer`s Health-Database <ref>https://www.dropbox.com/t/8LjP7cmulhr2x8Ty </ref></big> <big>1.3 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.4. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.5. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big><br /> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> <big>- John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref></big> <big>3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> <big>4. '''Prediabetes-Remission Research Network:'''</big> <small>Prof. Ing. MSc. Nailet Delgado; Prof. Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali Prof. SDU; MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey (Coordinator); Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark</small> ==References== <references /> [[Category:Prediabetes ]]Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> s5scbrkfyliqbq3rokqxtnsnulsqbf4 0 330700 2819658 2819114 2026-07-28T14:27:48Z Earthinators 3102737 2819658 wikitext text/x-wiki {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''An open, 5-week Wikiversity learning project bridging systems thinking, ecotoxicology, environmental policy, and community-driven circular solutions.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|330px|Free-ranging livestock ingesting single-use plastic waste near roadside vegetation, illustrating terrestrial pollution vectors and agricultural contamination.]] == 📌 Project Overview & Wikiversity Guidelines == Plastics are more than just waste. They are a lens to understand how our global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is an open, community-driven Wikiversity learning project. Guided by [[w:Systems thinking|systems thinking]], participants trace the complete plastic lifecycle, from oil extraction and chemical additives to ocean pollution, policy frameworks, and [[w:Environmental justice|environmental justice]]. Rather than just studying the crisis, you will actively create a solution. Step by step, you will design, launch, and document a real project in your school, workplace, or community. This could mean advocating for single-use plastic policy changes, creating a zero-waste refill campaign, or staging an environmental art project. In compliance with [[Wikiversity:Learning projects|Wikiversity Learning Project Guidelines]], this resource is 100% open-access, self-paced, and collaborative. All discussions, reading notes, and project logs are shared transparently on Wikiversity. <div style="background-color: #f0fdf4; border-left: 5px solid #16a34a; padding: 14px 18px; margin: 18px 0; border-radius: 0 8px 8px 0;"> <b style="color: #15803d; font-size: 1.05em;">💡 Core Educational Principles:</b> <ol style="margin-top: 6px; margin-bottom: 0;"> <li><b>Learning is Social:</b> Participants learn by posting reflections, sharing open-access resources, and giving peer feedback on Wikiversity discussion pages.</li> <li><b>Learning Should Lead to Action:</b> Participants apply course content directly to design and execute a tangible, locally-relevant plastic reduction initiative.</li> </ol> </div> == 📋 Project Information & Specifications == * '''School Affiliation:''' [[School:Environmental science|School of Environmental Science]] * '''Resource Classification:''' [[Wikiversity:Learning projects|Learning Project]] * '''Target Audience:''' Students, community activists, sustainability leaders, and concerned global citizens. No prior environmental science background required. * '''Format:''' 100% Asynchronous, open-access, self-paced OER learning. * '''Time Commitment:''' 3 to 4 hours per week for 5 weeks. * '''Required Tools:''' Notebook or digital log for plastic audits, camera/smartphone for field photos, and a free Wikiversity account. == 👥 Participant Roster & Cohort Sign-Up == Want to join this learning project? Add your Wikiversity username to the cohort list below! *(To sign up, edit this section and add <code># [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code> at the bottom)* # [[User:Earthinators|Earthinators Facilitator]] 14:26, 28 July 2026 (UTC) # ''(Sign up here to join the cohort!)'' == 🎯 Learning Outcomes == By the end of this course, participants will be able to: * '''Understand and Apply Systems Thinking:''' Use systems thinking to map the plastic lifecycle, global trade and waste flows, and environmental and health impacts. * '''Evaluate Governance Approaches:''' Discuss how feasible, effective, and fair different plastic policies and laws really are. * '''Launch a Local Action Project:''' Design and carry out a project targeting plastic pollution in your school, workplace, or community. * '''Reflect and Report:''' Analyze the results of your local project and write a 1 to 2 page reflection report sharing your outcomes and lessons learned. * '''Join a Global Community:''' Connect and collaborate with an international network of citizens taking action on plastic pollution. == 🗓️ 5-Week Action Journey & Comprehensive Syllabus == <div style="background: #f8fafc; border: 1px solid #cbd5e1; border-radius: 8px; padding: 12px 18px; margin-bottom: 20px;"> <b>🧭 Subpage Navigation:</b> [[/Week_1_Global_Plastic_System|Week 1: Global System]] • [[/Week_2_Ecological_and_Health_Impacts|Week 2: Impacts]] • [[/Week_3_Environmental_Policy_and_Governance|Week 3: Governance]] • [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives]] • [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Local Action]] </div> === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System - Oil Well to Household Waste]] === * '''Inquiry Question:''' Where does your plastic come from, and where does it go after disposal? * '''1.1 Petrochemical Origins and Industrial Expansion:''' Trace the transition from early synthetic resins like Parkesine and Bakelite to the post-World War II manufacturing boom that established modern mass disposable consumer culture. * '''1.2 Polymer Chemistry and Lifecycle Mapping:''' Examine how crude oil and natural gas refining feeds naphtha cracking into monomers, polymerization into resin pellets, and manufacturing into resin codes #1 PET through #7 OTHER. * '''1.3 Global Waste Leakage and Mass Balance Accounting:''' Quantify global municipal solid waste generation, mismanaged waste rates in developing infrastructure, and the breakdown of linear material throughput. * '''1.4 Environmental Justice and Global Waste Trade:''' Analyze environmental inequities in petrochemical refining corridors, international plastic waste dumping under the [[w:Basel Convention|Basel Convention]], and informal waste picker economies. * '''Weekly Milestone:''' Conduct a 24-hour personal plastic audit, log resin types (#1 to #7), and post your reflection on the <b>[[School talk:Environmental_science/Earthinators:_Plastic_Pollution_to_Circular_Solutions|Course Discussion Page]]</b> using <code>[[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code>. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Ecological & Human Health Impacts]] === * '''Inquiry Question:''' How do plastics and microplastics move through food chains and natural ecosystems? * '''2.1 Climate Systems and Petrochemical Carbon Emissions:''' Calculate life-cycle greenhouse gas emissions across crude extraction, refining, high-heat incineration, and solar breakdown of marine plastics releasing methane. * '''2.2 Wildlife Ecotoxicology and Chemical Leaching:''' Investigate physical entanglement hazards, digestive blockages in marine species, and chemical leaching of additives like phthalates, [[w:Bisphenol A|BPA]], and [[w:Per- and polyfluoroalkyl substances|PFAS]] disrupting endocrine systems. * '''2.3 Microplastics and Human Health Hazards:''' Differentiate between primary microbeads, secondary synthetic textile fibers, bioaccumulation in marine food webs, and emerging microplastic presence in human blood and lung tissues. * '''2.4 Ocean Contamination and Hydrodynamics:''' Track river export vectors delivering land-based debris into oceanic gyres, forming accumulation zones like the [[w:Great Pacific garbage patch|Great Pacific Garbage Patch]] and deep-sea floor sinks. * '''Weekly Milestone:''' Observe a local park, storm drain, or body of water, map out potential plastic contamination pathways, and share your notes on the discussion page. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Environmental Policy & Corporate Governance]] === * '''Inquiry Question:''' Who is responsible for plastic waste, and what policies actually work? * '''3.1 Systems Dynamics and International Treaties:''' Map system feedback loops, identify high-leverage intervention points, and analyze structural progress under the [[w:UN Plastics Treaty|UN Global Plastics Treaty]]. * '''3.2 Extended Producer Responsibility (EPR) Frameworks:''' Evaluate polluter-pays legislative models that mandate manufacturer financial accountability for post-consumer waste management and packaging redesign. * '''3.3 Recycling Thermodynamics and Chemical Reprocessing Limits:''' Examine thermodynamic limits of mechanical recycling, polymer chain degradation, downcycling realities, and energy claims surrounding chemical pyrolysis. * '''3.4 Legislative Instruments and Container Deposit Systems:''' Measure the real-world impact of single-use plastic bag bans, retail fees, packaging recycled-content mandates, and [[w:Container-deposit legislation|Container Deposit Systems (DRS)]] achieving over 90 percent recovery. * '''Weekly Milestone:''' Post a 1-paragraph proposal for your final local community action project on the discussion page for peer feedback. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Materials Innovation & Greenwashing Precautions]] === * '''Inquiry Question:''' How can we distinguish genuine sustainable innovations from corporate greenwashing? * '''4.1 Upcycling Safety and Structural Repurposing:''' Evaluate structural upcycling techniques like ecobricking, structural plastic lumber, and chemical leaching risks when re-exposed to outdoor heat and UV radiation. * '''4.2 Environmental Art and Public Consciousness:''' Study how visual metrics, debris installations, and community art campaigns shift cultural perceptions and pressure policy makers. * '''4.3 Bioplastics, Bio-Based Polymers, and Degradation Limits:''' Analyze plant-derived polymers like [[w:Bioplastics|PLA and PHA]], distinguishing high-temperature industrial composting requirements (>58C) from natural environmental breakdown limits. * '''4.4 Deconstructing Greenwashing and Deceptive Eco-Claims:''' Audit corporate distraction campaigns, industry front groups, and [[w:Greenwashing|ISO 14021 environmental labeling standards]] to spot misleading eco-claims. * '''Weekly Milestone:''' Perform a quick greenwashing audit on an eco-labeled consumer product and post your critique on the discussion page. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Circular Economies & Local Community Action]] === * '''Inquiry Question:''' How do we transition from linear plastic waste to circular community solutions? * '''5.1 Zero-Waste Systems and Reuse Logistics:''' Explore neighborhood refill networks, repair cafés, package-free stores, and returnable container logistics. * '''5.2 Citizen Science and Corporate Brand Audits:''' Master empirical litter sampling protocols and [[w:Break Free From Plastic|Break Free From Plastic]] brand audit methodologies to trace corporate polluters. * '''5.3 Post-Linear Economics and Material Sufficiency:''' Apply industrial ecology concepts, [[w:Degrowth|degrowth economics]], and Product-as-a-Service models to reduce total material throughput. * '''5.4 Capstone Execution and Analytical Reflection:''' Finalize your community intervention, compile field data or photos, and write a 1 to 2 page reflection report analyzing project outcomes and systems leverage. * '''Final Milestone:''' Post your project summary, campaign materials, or reflection report on the course discussion page to complete the course. == 🏆 Assessment & Wikiversity Userbox Badge == Participants who complete all 5 weekly milestones and post their final 1 to 2 page reflection report on the course discussion page earn the '''Earthinators Circular Action Badge''' to display on their Wikiversity User page! <div style="background: #f1f5f9; border: 1px solid #cbd5e1; padding: 10px 14px; border-radius: 8px; width: fit-content;"> <code>{{User Wikiversity/Earthinators}}</code> — <b>Earthinators Circular Action Graduate</b> </div> == 📖 Open Educational Resources (OER) & Wikimedia Integration == === 🌿 Earthinators Initiative === * [[c:Category:Earthinators|Earthinators Media Gallery on Wikimedia Commons]] * [[d:Q103824308|Earthinators Knowledge Item on Wikidata]] === 🌊 Sister Wikimedia Projects & Portals === * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media & Videos on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] * [[b:Subject:Environmental science|Environmental Science Books on Wikibooks]] * [[q:Rachel Carson|Rachel Carson Quotes on Wikiquote]] <div style="border: 1px solid #cbd5e1; background: #f8fafc; padding: 12px 16px; border-radius: 8px; font-size: 0.88em; margin-top: 24px;"> <b>🔓 Open Educational Resource (OER) License:</b><br/> In full accordance with Wikiversity policy, all educational content on this page is licensed under the <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0)</a>. Feel free to reuse, adapt, and remix for non-commercial or commercial educational purposes. </div> [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] h74yc6nhumgquq3g8y6gucvtz45rxaq 2819665 2819658 2026-07-28T15:06:22Z Earthinators 3102737 2819665 wikitext text/x-wiki {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''An open, 5-week Wikiversity learning project bridging systems thinking, ecotoxicology, environmental policy, and community-driven circular solutions.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|330px|Free-ranging livestock ingesting single-use plastic waste near roadside vegetation, illustrating terrestrial pollution vectors and agricultural contamination.]] == 📌 Project Overview & Wikiversity Guidelines == Plastics are more than just waste. They are a lens to understand how our global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is an open, community-driven Wikiversity learning project. Guided by [[w:Systems thinking|systems thinking]], participants trace the complete plastic lifecycle, from oil extraction and chemical additives to ocean pollution, policy frameworks, and [[w:Environmental justice|environmental justice]]. Rather than just studying the crisis, you will actively create a solution. Step by step, you will design, launch, and document a real project in your school, workplace, or community. This could mean advocating for single-use plastic policy changes, creating a zero-waste refill campaign, or staging an environmental art project. In compliance with [[Wikiversity:Learning projects|Wikiversity Learning Project Guidelines]], this resource is 100% open-access, self-paced, and collaborative. All discussions, reading notes, and project logs are shared transparently on Wikiversity. <div style="background-color: #f0fdf4; border-left: 5px solid #16a34a; padding: 14px 18px; margin: 18px 0; border-radius: 0 8px 8px 0;"> <b style="color: #15803d; font-size: 1.05em;">💡 Core Educational Principles:</b> <ol style="margin-top: 6px; margin-bottom: 0;"> <li><b>Learning is Social:</b> Participants learn by posting reflections, sharing open-access resources, and giving peer feedback on Wikiversity discussion pages.</li> <li><b>Learning Should Lead to Action:</b> Participants apply course content directly to design and execute a tangible, locally-relevant plastic reduction initiative.</li> </ol> </div> == 📋 Project Information & Specifications == * '''School Affiliation:''' [[School:Environmental science|School of Environmental Science]] * '''Resource Classification:''' [[Wikiversity:Learning projects|Learning Project]] * '''Target Audience:''' Students, community activists, sustainability leaders, and concerned global citizens. No prior environmental science background required. * '''Format:''' 100% Asynchronous, open-access, self-paced OER learning. * '''Time Commitment:''' 3 to 4 hours per week for 5 weeks. * '''Required Tools:''' Notebook or digital log for plastic audits, camera/smartphone for field photos, and a free Wikiversity account. == 👥 Participant Roster & Cohort Sign-Up == Want to join this learning project? Add your Wikiversity username to the cohort list below! *(To sign up, edit this section and add <code># [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code> at the bottom)* # [[User:Earthinators|Earthinators Facilitator]] 14:26, 28 July 2026 (UTC) # ''(Sign up here to join the cohort!)'' == 🎯 Learning Outcomes == By the end of this course, participants will be able to: * '''Understand and Apply Systems Thinking:''' Use systems thinking to map the plastic lifecycle, global trade and waste flows, and environmental and health impacts. * '''Evaluate Governance Approaches:''' Discuss how feasible, effective, and fair different plastic policies and laws really are. * '''Launch a Local Action Project:''' Design and carry out a project targeting plastic pollution in your school, workplace, or community. * '''Reflect and Report:''' Analyze the results of your local project and write a 1 to 2 page reflection report sharing your outcomes and lessons learned. * '''Join a Global Community:''' Connect and collaborate with an international network of citizens taking action on plastic pollution. == 🗓️ 5-Week Action Journey & Comprehensive Syllabus == <div style="background: #f8fafc; border: 1px solid #cbd5e1; border-radius: 8px; padding: 12px 18px; margin-bottom: 20px;"> <b>🧭 Subpage Navigation:</b> [[/Week_1_Global_Plastic_System|Week 1: Global System]] • [[/Week_2_Ecological_and_Health_Impacts|Week 2: Impacts]] • [[/Week_3_Environmental_Policy_and_Governance|Week 3: Governance]] • [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives]] • [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Local Action]] </div> === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System - Oil Well to Household Waste]] === * '''Inquiry Question:''' Where does your plastic come from, and where does it go after disposal? * '''1.1 Petrochemical Origins and Industrial Expansion:''' Trace the transition from early synthetic resins like Parkesine and Bakelite to the post-World War II manufacturing boom that established modern mass disposable consumer culture. * '''1.2 Polymer Chemistry and Lifecycle Mapping:''' Examine how crude oil and natural gas refining feeds naphtha cracking into monomers, polymerization into resin pellets, and manufacturing into resin codes #1 PET through #7 OTHER. * '''1.3 Global Waste Leakage and Mass Balance Accounting:''' Quantify global municipal solid waste generation, mismanaged waste rates in developing infrastructure, and the breakdown of linear material throughput. * '''1.4 Environmental Justice and Global Waste Trade:''' Analyze environmental inequities in petrochemical refining corridors, international plastic waste dumping under the [[w:Basel Convention|Basel Convention]], and informal waste picker economies. * '''Weekly Milestone:''' Conduct a 24-hour personal plastic audit, log resin types (#1 to #7), and post your reflection on the <b>[[School talk:Environmental_science/Earthinators:_Plastic_Pollution_to_Circular_Solutions|Course Discussion Page]]</b> using <code>[[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code>. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Ecological & Human Health Impacts]] === * '''Inquiry Question:''' How do plastics and microplastics move through food chains and natural ecosystems? * '''2.1 Climate Systems and Petrochemical Carbon Emissions:''' Calculate life-cycle greenhouse gas emissions across crude extraction, refining, high-heat incineration, and solar breakdown of marine plastics releasing methane. * '''2.2 Wildlife Ecotoxicology and Chemical Leaching:''' Investigate physical entanglement hazards, digestive blockages in marine species, and chemical leaching of additives like phthalates, [[w:Bisphenol A|BPA]], and [[w:Per- and polyfluoroalkyl substances|PFAS]] disrupting endocrine systems. * '''2.3 Microplastics and Human Health Hazards:''' Differentiate between primary microbeads, secondary synthetic textile fibers, bioaccumulation in marine food webs, and emerging microplastic presence in human blood and lung tissues. * '''2.4 Ocean Contamination and Hydrodynamics:''' Track river export vectors delivering land-based debris into oceanic gyres, forming accumulation zones like the [[w:Great Pacific garbage patch|Great Pacific Garbage Patch]] and deep-sea floor sinks. * '''Weekly Milestone:''' Observe a local park, storm drain, or body of water, map out potential plastic contamination pathways, and share your notes on the discussion page. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Environmental Policy & Corporate Governance]] === * '''Inquiry Question:''' Who is responsible for plastic waste, and what policies actually work? * '''3.1 Systems Dynamics and International Treaties:''' Map system feedback loops, identify high-leverage intervention points, and analyze structural progress under the [[w:UN Plastics Treaty|UN Global Plastics Treaty]]. * '''3.2 Extended Producer Responsibility (EPR) Frameworks:''' Evaluate polluter-pays legislative models that mandate manufacturer financial accountability for post-consumer waste management and packaging redesign. * '''3.3 Recycling Thermodynamics and Chemical Reprocessing Limits:''' Examine thermodynamic limits of mechanical recycling, polymer chain degradation, downcycling realities, and energy claims surrounding chemical pyrolysis. * '''3.4 Legislative Instruments and Container Deposit Systems:''' Measure the real-world impact of single-use plastic bag bans, retail fees, packaging recycled-content mandates, and [[w:Container-deposit legislation|Container Deposit Systems (DRS)]] achieving over 90 percent recovery. * '''Weekly Milestone:''' Post a 1-paragraph proposal for your final local community action project on the discussion page for peer feedback. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Materials Innovation & Greenwashing Precautions]] === * '''Inquiry Question:''' How can we distinguish genuine sustainable innovations from corporate greenwashing? * '''4.1 Upcycling Safety and Structural Repurposing:''' Evaluate structural upcycling techniques like ecobricking, structural plastic lumber, and chemical leaching risks when re-exposed to outdoor heat and UV radiation. * '''4.2 Environmental Art and Public Consciousness:''' Study how visual metrics, debris installations, and community art campaigns shift cultural perceptions and pressure policy makers. * '''4.3 Bioplastics, Bio-Based Polymers, and Degradation Limits:''' Analyze plant-derived polymers like [[w:Bioplastics|PLA and PHA]], distinguishing high-temperature industrial composting requirements (>58C) from natural environmental breakdown limits. * '''4.4 Deconstructing Greenwashing and Deceptive Eco-Claims:''' Audit corporate distraction campaigns, industry front groups, and [[w:Greenwashing|ISO 14021 environmental labeling standards]] to spot misleading eco-claims. * '''Weekly Milestone:''' Perform a quick greenwashing audit on an eco-labeled consumer product and post your critique on the discussion page. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Circular Economies & Local Community Action]] === * '''Inquiry Question:''' How do we transition from linear plastic waste to circular community solutions? * '''5.1 Zero-Waste Systems and Reuse Logistics:''' Explore neighborhood refill networks, repair cafés, package-free stores, and returnable container logistics. * '''5.2 Citizen Science and Corporate Brand Audits:''' Master empirical litter sampling protocols and [[w:Break Free From Plastic|Break Free From Plastic]] brand audit methodologies to trace corporate polluters. * '''5.3 Post-Linear Economics and Material Sufficiency:''' Apply industrial ecology concepts, [[w:Degrowth|degrowth economics]], and Product-as-a-Service models to reduce total material throughput. * '''5.4 Capstone Execution and Analytical Reflection:''' Finalize your community intervention, compile field data or photos, and write a 1 to 2 page reflection report analyzing project outcomes and systems leverage. * '''Final Milestone:''' Post your project summary, campaign materials, or reflection report on the course discussion page to complete the course. == 🏆 Assessment & Wikiversity Userbox Badge == Participants who complete all 5 weekly milestones and post their final 1 to 2 page reflection report on the course discussion page earn the '''Earthinators Circular Action Badge''' to display on their Wikiversity User page! <div style="background: #f1f5f9; border: 1px solid #cbd5e1; padding: 10px 14px; border-radius: 8px; width: fit-content;"> <code>{{User Wikiversity/Earthinators}}</code> — <b>Earthinators Circular Action Graduate</b> </div> == 📖 Open Educational Resources (OER) & Wikimedia Integration == === 🌿 Earthinators Initiative === * [[c:Category:Earthinators|Earthinators Media Gallery on Wikimedia Commons]] * [[d:Q103824308|Earthinators Knowledge Item on Wikidata]] === 🌊 Sister Wikimedia Projects & Portals === * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media & Videos on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] * [[b:Subject:Environmental sciences|Environmental Science Books on Wikibooks]] * [[q:Rachel Carson|Rachel Carson Quotes on Wikiquote]] <div style="border: 1px solid #cbd5e1; background: #f8fafc; padding: 12px 16px; border-radius: 8px; font-size: 0.88em; margin-top: 24px;"> <b>🔓 Open Educational Resource (OER) License:</b><br/> In full accordance with Wikiversity policy, all educational content on this page is licensed under the <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0)</a>. Feel free to reuse, adapt, and remix for non-commercial or commercial educational purposes. </div> [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] 3b0iia2vuw0hnyvjizlgnschamii5ze 2819667 2819665 2026-07-28T15:07:39Z Atcovi 276019 hallmarks of AI-generated content are observed, so {{AI-generated}} should be placed 2819667 wikitext text/x-wiki {{AI-generated}} {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''An open, 5-week Wikiversity learning project bridging systems thinking, ecotoxicology, environmental policy, and community-driven circular solutions.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|330px|Free-ranging livestock ingesting single-use plastic waste near roadside vegetation, illustrating terrestrial pollution vectors and agricultural contamination.]] == 📌 Project Overview & Wikiversity Guidelines == Plastics are more than just waste. They are a lens to understand how our global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is an open, community-driven Wikiversity learning project. Guided by [[w:Systems thinking|systems thinking]], participants trace the complete plastic lifecycle, from oil extraction and chemical additives to ocean pollution, policy frameworks, and [[w:Environmental justice|environmental justice]]. Rather than just studying the crisis, you will actively create a solution. Step by step, you will design, launch, and document a real project in your school, workplace, or community. This could mean advocating for single-use plastic policy changes, creating a zero-waste refill campaign, or staging an environmental art project. In compliance with [[Wikiversity:Learning projects|Wikiversity Learning Project Guidelines]], this resource is 100% open-access, self-paced, and collaborative. All discussions, reading notes, and project logs are shared transparently on Wikiversity. <div style="background-color: #f0fdf4; border-left: 5px solid #16a34a; padding: 14px 18px; margin: 18px 0; border-radius: 0 8px 8px 0;"> <b style="color: #15803d; font-size: 1.05em;">💡 Core Educational Principles:</b> <ol style="margin-top: 6px; margin-bottom: 0;"> <li><b>Learning is Social:</b> Participants learn by posting reflections, sharing open-access resources, and giving peer feedback on Wikiversity discussion pages.</li> <li><b>Learning Should Lead to Action:</b> Participants apply course content directly to design and execute a tangible, locally-relevant plastic reduction initiative.</li> </ol> </div> == 📋 Project Information & Specifications == * '''School Affiliation:''' [[School:Environmental science|School of Environmental Science]] * '''Resource Classification:''' [[Wikiversity:Learning projects|Learning Project]] * '''Target Audience:''' Students, community activists, sustainability leaders, and concerned global citizens. No prior environmental science background required. * '''Format:''' 100% Asynchronous, open-access, self-paced OER learning. * '''Time Commitment:''' 3 to 4 hours per week for 5 weeks. * '''Required Tools:''' Notebook or digital log for plastic audits, camera/smartphone for field photos, and a free Wikiversity account. == 👥 Participant Roster & Cohort Sign-Up == Want to join this learning project? Add your Wikiversity username to the cohort list below! *(To sign up, edit this section and add <code># [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code> at the bottom)* # [[User:Earthinators|Earthinators Facilitator]] 14:26, 28 July 2026 (UTC) # ''(Sign up here to join the cohort!)'' == 🎯 Learning Outcomes == By the end of this course, participants will be able to: * '''Understand and Apply Systems Thinking:''' Use systems thinking to map the plastic lifecycle, global trade and waste flows, and environmental and health impacts. * '''Evaluate Governance Approaches:''' Discuss how feasible, effective, and fair different plastic policies and laws really are. * '''Launch a Local Action Project:''' Design and carry out a project targeting plastic pollution in your school, workplace, or community. * '''Reflect and Report:''' Analyze the results of your local project and write a 1 to 2 page reflection report sharing your outcomes and lessons learned. * '''Join a Global Community:''' Connect and collaborate with an international network of citizens taking action on plastic pollution. == 🗓️ 5-Week Action Journey & Comprehensive Syllabus == <div style="background: #f8fafc; border: 1px solid #cbd5e1; border-radius: 8px; padding: 12px 18px; margin-bottom: 20px;"> <b>🧭 Subpage Navigation:</b> [[/Week_1_Global_Plastic_System|Week 1: Global System]] • [[/Week_2_Ecological_and_Health_Impacts|Week 2: Impacts]] • [[/Week_3_Environmental_Policy_and_Governance|Week 3: Governance]] • [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives]] • [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Local Action]] </div> === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System - Oil Well to Household Waste]] === * '''Inquiry Question:''' Where does your plastic come from, and where does it go after disposal? * '''1.1 Petrochemical Origins and Industrial Expansion:''' Trace the transition from early synthetic resins like Parkesine and Bakelite to the post-World War II manufacturing boom that established modern mass disposable consumer culture. * '''1.2 Polymer Chemistry and Lifecycle Mapping:''' Examine how crude oil and natural gas refining feeds naphtha cracking into monomers, polymerization into resin pellets, and manufacturing into resin codes #1 PET through #7 OTHER. * '''1.3 Global Waste Leakage and Mass Balance Accounting:''' Quantify global municipal solid waste generation, mismanaged waste rates in developing infrastructure, and the breakdown of linear material throughput. * '''1.4 Environmental Justice and Global Waste Trade:''' Analyze environmental inequities in petrochemical refining corridors, international plastic waste dumping under the [[w:Basel Convention|Basel Convention]], and informal waste picker economies. * '''Weekly Milestone:''' Conduct a 24-hour personal plastic audit, log resin types (#1 to #7), and post your reflection on the <b>[[School talk:Environmental_science/Earthinators:_Plastic_Pollution_to_Circular_Solutions|Course Discussion Page]]</b> using <code>[[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code>. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Ecological & Human Health Impacts]] === * '''Inquiry Question:''' How do plastics and microplastics move through food chains and natural ecosystems? * '''2.1 Climate Systems and Petrochemical Carbon Emissions:''' Calculate life-cycle greenhouse gas emissions across crude extraction, refining, high-heat incineration, and solar breakdown of marine plastics releasing methane. * '''2.2 Wildlife Ecotoxicology and Chemical Leaching:''' Investigate physical entanglement hazards, digestive blockages in marine species, and chemical leaching of additives like phthalates, [[w:Bisphenol A|BPA]], and [[w:Per- and polyfluoroalkyl substances|PFAS]] disrupting endocrine systems. * '''2.3 Microplastics and Human Health Hazards:''' Differentiate between primary microbeads, secondary synthetic textile fibers, bioaccumulation in marine food webs, and emerging microplastic presence in human blood and lung tissues. * '''2.4 Ocean Contamination and Hydrodynamics:''' Track river export vectors delivering land-based debris into oceanic gyres, forming accumulation zones like the [[w:Great Pacific garbage patch|Great Pacific Garbage Patch]] and deep-sea floor sinks. * '''Weekly Milestone:''' Observe a local park, storm drain, or body of water, map out potential plastic contamination pathways, and share your notes on the discussion page. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Environmental Policy & Corporate Governance]] === * '''Inquiry Question:''' Who is responsible for plastic waste, and what policies actually work? * '''3.1 Systems Dynamics and International Treaties:''' Map system feedback loops, identify high-leverage intervention points, and analyze structural progress under the [[w:UN Plastics Treaty|UN Global Plastics Treaty]]. * '''3.2 Extended Producer Responsibility (EPR) Frameworks:''' Evaluate polluter-pays legislative models that mandate manufacturer financial accountability for post-consumer waste management and packaging redesign. * '''3.3 Recycling Thermodynamics and Chemical Reprocessing Limits:''' Examine thermodynamic limits of mechanical recycling, polymer chain degradation, downcycling realities, and energy claims surrounding chemical pyrolysis. * '''3.4 Legislative Instruments and Container Deposit Systems:''' Measure the real-world impact of single-use plastic bag bans, retail fees, packaging recycled-content mandates, and [[w:Container-deposit legislation|Container Deposit Systems (DRS)]] achieving over 90 percent recovery. * '''Weekly Milestone:''' Post a 1-paragraph proposal for your final local community action project on the discussion page for peer feedback. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Materials Innovation & Greenwashing Precautions]] === * '''Inquiry Question:''' How can we distinguish genuine sustainable innovations from corporate greenwashing? * '''4.1 Upcycling Safety and Structural Repurposing:''' Evaluate structural upcycling techniques like ecobricking, structural plastic lumber, and chemical leaching risks when re-exposed to outdoor heat and UV radiation. * '''4.2 Environmental Art and Public Consciousness:''' Study how visual metrics, debris installations, and community art campaigns shift cultural perceptions and pressure policy makers. * '''4.3 Bioplastics, Bio-Based Polymers, and Degradation Limits:''' Analyze plant-derived polymers like [[w:Bioplastics|PLA and PHA]], distinguishing high-temperature industrial composting requirements (>58C) from natural environmental breakdown limits. * '''4.4 Deconstructing Greenwashing and Deceptive Eco-Claims:''' Audit corporate distraction campaigns, industry front groups, and [[w:Greenwashing|ISO 14021 environmental labeling standards]] to spot misleading eco-claims. * '''Weekly Milestone:''' Perform a quick greenwashing audit on an eco-labeled consumer product and post your critique on the discussion page. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Circular Economies & Local Community Action]] === * '''Inquiry Question:''' How do we transition from linear plastic waste to circular community solutions? * '''5.1 Zero-Waste Systems and Reuse Logistics:''' Explore neighborhood refill networks, repair cafés, package-free stores, and returnable container logistics. * '''5.2 Citizen Science and Corporate Brand Audits:''' Master empirical litter sampling protocols and [[w:Break Free From Plastic|Break Free From Plastic]] brand audit methodologies to trace corporate polluters. * '''5.3 Post-Linear Economics and Material Sufficiency:''' Apply industrial ecology concepts, [[w:Degrowth|degrowth economics]], and Product-as-a-Service models to reduce total material throughput. * '''5.4 Capstone Execution and Analytical Reflection:''' Finalize your community intervention, compile field data or photos, and write a 1 to 2 page reflection report analyzing project outcomes and systems leverage. * '''Final Milestone:''' Post your project summary, campaign materials, or reflection report on the course discussion page to complete the course. == 🏆 Assessment & Wikiversity Userbox Badge == Participants who complete all 5 weekly milestones and post their final 1 to 2 page reflection report on the course discussion page earn the '''Earthinators Circular Action Badge''' to display on their Wikiversity User page! <div style="background: #f1f5f9; border: 1px solid #cbd5e1; padding: 10px 14px; border-radius: 8px; width: fit-content;"> <code>{{User Wikiversity/Earthinators}}</code> — <b>Earthinators Circular Action Graduate</b> </div> == 📖 Open Educational Resources (OER) & Wikimedia Integration == === 🌿 Earthinators Initiative === * [[c:Category:Earthinators|Earthinators Media Gallery on Wikimedia Commons]] * [[d:Q103824308|Earthinators Knowledge Item on Wikidata]] === 🌊 Sister Wikimedia Projects & Portals === * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media & Videos on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] * [[b:Subject:Environmental sciences|Environmental Science Books on Wikibooks]] * [[q:Rachel Carson|Rachel Carson Quotes on Wikiquote]] <div style="border: 1px solid #cbd5e1; background: #f8fafc; padding: 12px 16px; border-radius: 8px; font-size: 0.88em; margin-top: 24px;"> <b>🔓 Open Educational Resource (OER) License:</b><br/> In full accordance with Wikiversity policy, all educational content on this page is licensed under the <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0)</a>. Feel free to reuse, adapt, and remix for non-commercial or commercial educational purposes. </div> [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] nw7xgpge8wzlb4b994ci8h306482snb 2819670 2819667 2026-07-28T15:12:03Z Atcovi 276019 Atcovi moved page [[School:Environmental science/Earthinators: Plastic Pollution to Circular Solutions]] to [[Environmental science/Earthinators: Plastic Pollution to Circular Solutions]] without leaving a redirect: should be under mainspace, not School 2819667 wikitext text/x-wiki {{AI-generated}} {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''An open, 5-week Wikiversity learning project bridging systems thinking, ecotoxicology, environmental policy, and community-driven circular solutions.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|330px|Free-ranging livestock ingesting single-use plastic waste near roadside vegetation, illustrating terrestrial pollution vectors and agricultural contamination.]] == 📌 Project Overview & Wikiversity Guidelines == Plastics are more than just waste. They are a lens to understand how our global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is an open, community-driven Wikiversity learning project. Guided by [[w:Systems thinking|systems thinking]], participants trace the complete plastic lifecycle, from oil extraction and chemical additives to ocean pollution, policy frameworks, and [[w:Environmental justice|environmental justice]]. Rather than just studying the crisis, you will actively create a solution. Step by step, you will design, launch, and document a real project in your school, workplace, or community. This could mean advocating for single-use plastic policy changes, creating a zero-waste refill campaign, or staging an environmental art project. In compliance with [[Wikiversity:Learning projects|Wikiversity Learning Project Guidelines]], this resource is 100% open-access, self-paced, and collaborative. All discussions, reading notes, and project logs are shared transparently on Wikiversity. <div style="background-color: #f0fdf4; border-left: 5px solid #16a34a; padding: 14px 18px; margin: 18px 0; border-radius: 0 8px 8px 0;"> <b style="color: #15803d; font-size: 1.05em;">💡 Core Educational Principles:</b> <ol style="margin-top: 6px; margin-bottom: 0;"> <li><b>Learning is Social:</b> Participants learn by posting reflections, sharing open-access resources, and giving peer feedback on Wikiversity discussion pages.</li> <li><b>Learning Should Lead to Action:</b> Participants apply course content directly to design and execute a tangible, locally-relevant plastic reduction initiative.</li> </ol> </div> == 📋 Project Information & Specifications == * '''School Affiliation:''' [[School:Environmental science|School of Environmental Science]] * '''Resource Classification:''' [[Wikiversity:Learning projects|Learning Project]] * '''Target Audience:''' Students, community activists, sustainability leaders, and concerned global citizens. No prior environmental science background required. * '''Format:''' 100% Asynchronous, open-access, self-paced OER learning. * '''Time Commitment:''' 3 to 4 hours per week for 5 weeks. * '''Required Tools:''' Notebook or digital log for plastic audits, camera/smartphone for field photos, and a free Wikiversity account. == 👥 Participant Roster & Cohort Sign-Up == Want to join this learning project? Add your Wikiversity username to the cohort list below! *(To sign up, edit this section and add <code># [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code> at the bottom)* # [[User:Earthinators|Earthinators Facilitator]] 14:26, 28 July 2026 (UTC) # ''(Sign up here to join the cohort!)'' == 🎯 Learning Outcomes == By the end of this course, participants will be able to: * '''Understand and Apply Systems Thinking:''' Use systems thinking to map the plastic lifecycle, global trade and waste flows, and environmental and health impacts. * '''Evaluate Governance Approaches:''' Discuss how feasible, effective, and fair different plastic policies and laws really are. * '''Launch a Local Action Project:''' Design and carry out a project targeting plastic pollution in your school, workplace, or community. * '''Reflect and Report:''' Analyze the results of your local project and write a 1 to 2 page reflection report sharing your outcomes and lessons learned. * '''Join a Global Community:''' Connect and collaborate with an international network of citizens taking action on plastic pollution. == 🗓️ 5-Week Action Journey & Comprehensive Syllabus == <div style="background: #f8fafc; border: 1px solid #cbd5e1; border-radius: 8px; padding: 12px 18px; margin-bottom: 20px;"> <b>🧭 Subpage Navigation:</b> [[/Week_1_Global_Plastic_System|Week 1: Global System]] • [[/Week_2_Ecological_and_Health_Impacts|Week 2: Impacts]] • [[/Week_3_Environmental_Policy_and_Governance|Week 3: Governance]] • [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives]] • [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Local Action]] </div> === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System - Oil Well to Household Waste]] === * '''Inquiry Question:''' Where does your plastic come from, and where does it go after disposal? * '''1.1 Petrochemical Origins and Industrial Expansion:''' Trace the transition from early synthetic resins like Parkesine and Bakelite to the post-World War II manufacturing boom that established modern mass disposable consumer culture. * '''1.2 Polymer Chemistry and Lifecycle Mapping:''' Examine how crude oil and natural gas refining feeds naphtha cracking into monomers, polymerization into resin pellets, and manufacturing into resin codes #1 PET through #7 OTHER. * '''1.3 Global Waste Leakage and Mass Balance Accounting:''' Quantify global municipal solid waste generation, mismanaged waste rates in developing infrastructure, and the breakdown of linear material throughput. * '''1.4 Environmental Justice and Global Waste Trade:''' Analyze environmental inequities in petrochemical refining corridors, international plastic waste dumping under the [[w:Basel Convention|Basel Convention]], and informal waste picker economies. * '''Weekly Milestone:''' Conduct a 24-hour personal plastic audit, log resin types (#1 to #7), and post your reflection on the <b>[[School talk:Environmental_science/Earthinators:_Plastic_Pollution_to_Circular_Solutions|Course Discussion Page]]</b> using <code>[[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code>. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Ecological & Human Health Impacts]] === * '''Inquiry Question:''' How do plastics and microplastics move through food chains and natural ecosystems? * '''2.1 Climate Systems and Petrochemical Carbon Emissions:''' Calculate life-cycle greenhouse gas emissions across crude extraction, refining, high-heat incineration, and solar breakdown of marine plastics releasing methane. * '''2.2 Wildlife Ecotoxicology and Chemical Leaching:''' Investigate physical entanglement hazards, digestive blockages in marine species, and chemical leaching of additives like phthalates, [[w:Bisphenol A|BPA]], and [[w:Per- and polyfluoroalkyl substances|PFAS]] disrupting endocrine systems. * '''2.3 Microplastics and Human Health Hazards:''' Differentiate between primary microbeads, secondary synthetic textile fibers, bioaccumulation in marine food webs, and emerging microplastic presence in human blood and lung tissues. * '''2.4 Ocean Contamination and Hydrodynamics:''' Track river export vectors delivering land-based debris into oceanic gyres, forming accumulation zones like the [[w:Great Pacific garbage patch|Great Pacific Garbage Patch]] and deep-sea floor sinks. * '''Weekly Milestone:''' Observe a local park, storm drain, or body of water, map out potential plastic contamination pathways, and share your notes on the discussion page. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Environmental Policy & Corporate Governance]] === * '''Inquiry Question:''' Who is responsible for plastic waste, and what policies actually work? * '''3.1 Systems Dynamics and International Treaties:''' Map system feedback loops, identify high-leverage intervention points, and analyze structural progress under the [[w:UN Plastics Treaty|UN Global Plastics Treaty]]. * '''3.2 Extended Producer Responsibility (EPR) Frameworks:''' Evaluate polluter-pays legislative models that mandate manufacturer financial accountability for post-consumer waste management and packaging redesign. * '''3.3 Recycling Thermodynamics and Chemical Reprocessing Limits:''' Examine thermodynamic limits of mechanical recycling, polymer chain degradation, downcycling realities, and energy claims surrounding chemical pyrolysis. * '''3.4 Legislative Instruments and Container Deposit Systems:''' Measure the real-world impact of single-use plastic bag bans, retail fees, packaging recycled-content mandates, and [[w:Container-deposit legislation|Container Deposit Systems (DRS)]] achieving over 90 percent recovery. * '''Weekly Milestone:''' Post a 1-paragraph proposal for your final local community action project on the discussion page for peer feedback. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Materials Innovation & Greenwashing Precautions]] === * '''Inquiry Question:''' How can we distinguish genuine sustainable innovations from corporate greenwashing? * '''4.1 Upcycling Safety and Structural Repurposing:''' Evaluate structural upcycling techniques like ecobricking, structural plastic lumber, and chemical leaching risks when re-exposed to outdoor heat and UV radiation. * '''4.2 Environmental Art and Public Consciousness:''' Study how visual metrics, debris installations, and community art campaigns shift cultural perceptions and pressure policy makers. * '''4.3 Bioplastics, Bio-Based Polymers, and Degradation Limits:''' Analyze plant-derived polymers like [[w:Bioplastics|PLA and PHA]], distinguishing high-temperature industrial composting requirements (>58C) from natural environmental breakdown limits. * '''4.4 Deconstructing Greenwashing and Deceptive Eco-Claims:''' Audit corporate distraction campaigns, industry front groups, and [[w:Greenwashing|ISO 14021 environmental labeling standards]] to spot misleading eco-claims. * '''Weekly Milestone:''' Perform a quick greenwashing audit on an eco-labeled consumer product and post your critique on the discussion page. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Circular Economies & Local Community Action]] === * '''Inquiry Question:''' How do we transition from linear plastic waste to circular community solutions? * '''5.1 Zero-Waste Systems and Reuse Logistics:''' Explore neighborhood refill networks, repair cafés, package-free stores, and returnable container logistics. * '''5.2 Citizen Science and Corporate Brand Audits:''' Master empirical litter sampling protocols and [[w:Break Free From Plastic|Break Free From Plastic]] brand audit methodologies to trace corporate polluters. * '''5.3 Post-Linear Economics and Material Sufficiency:''' Apply industrial ecology concepts, [[w:Degrowth|degrowth economics]], and Product-as-a-Service models to reduce total material throughput. * '''5.4 Capstone Execution and Analytical Reflection:''' Finalize your community intervention, compile field data or photos, and write a 1 to 2 page reflection report analyzing project outcomes and systems leverage. * '''Final Milestone:''' Post your project summary, campaign materials, or reflection report on the course discussion page to complete the course. == 🏆 Assessment & Wikiversity Userbox Badge == Participants who complete all 5 weekly milestones and post their final 1 to 2 page reflection report on the course discussion page earn the '''Earthinators Circular Action Badge''' to display on their Wikiversity User page! <div style="background: #f1f5f9; border: 1px solid #cbd5e1; padding: 10px 14px; border-radius: 8px; width: fit-content;"> <code>{{User Wikiversity/Earthinators}}</code> — <b>Earthinators Circular Action Graduate</b> </div> == 📖 Open Educational Resources (OER) & Wikimedia Integration == === 🌿 Earthinators Initiative === * [[c:Category:Earthinators|Earthinators Media Gallery on Wikimedia Commons]] * [[d:Q103824308|Earthinators Knowledge Item on Wikidata]] === 🌊 Sister Wikimedia Projects & Portals === * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media & Videos on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] * [[b:Subject:Environmental sciences|Environmental Science Books on Wikibooks]] * [[q:Rachel Carson|Rachel Carson Quotes on Wikiquote]] <div style="border: 1px solid #cbd5e1; background: #f8fafc; padding: 12px 16px; border-radius: 8px; font-size: 0.88em; margin-top: 24px;"> <b>🔓 Open Educational Resource (OER) License:</b><br/> In full accordance with Wikiversity policy, all educational content on this page is licensed under the <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0)</a>. Feel free to reuse, adapt, and remix for non-commercial or commercial educational purposes. </div> [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] nw7xgpge8wzlb4b994ci8h306482snb 2819672 2819670 2026-07-28T15:12:52Z Earthinators 3102737 Earthinators moved page [[Environmental science/Earthinators: Plastic Pollution to Circular Solutions]] to [[School:Environmental science/Earthinators: Plastic Pollution to Circular Solutions]]: Revert 2819667 wikitext text/x-wiki {{AI-generated}} {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''An open, 5-week Wikiversity learning project bridging systems thinking, ecotoxicology, environmental policy, and community-driven circular solutions.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|330px|Free-ranging livestock ingesting single-use plastic waste near roadside vegetation, illustrating terrestrial pollution vectors and agricultural contamination.]] == 📌 Project Overview & Wikiversity Guidelines == Plastics are more than just waste. They are a lens to understand how our global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is an open, community-driven Wikiversity learning project. Guided by [[w:Systems thinking|systems thinking]], participants trace the complete plastic lifecycle, from oil extraction and chemical additives to ocean pollution, policy frameworks, and [[w:Environmental justice|environmental justice]]. Rather than just studying the crisis, you will actively create a solution. Step by step, you will design, launch, and document a real project in your school, workplace, or community. This could mean advocating for single-use plastic policy changes, creating a zero-waste refill campaign, or staging an environmental art project. In compliance with [[Wikiversity:Learning projects|Wikiversity Learning Project Guidelines]], this resource is 100% open-access, self-paced, and collaborative. All discussions, reading notes, and project logs are shared transparently on Wikiversity. <div style="background-color: #f0fdf4; border-left: 5px solid #16a34a; padding: 14px 18px; margin: 18px 0; border-radius: 0 8px 8px 0;"> <b style="color: #15803d; font-size: 1.05em;">💡 Core Educational Principles:</b> <ol style="margin-top: 6px; margin-bottom: 0;"> <li><b>Learning is Social:</b> Participants learn by posting reflections, sharing open-access resources, and giving peer feedback on Wikiversity discussion pages.</li> <li><b>Learning Should Lead to Action:</b> Participants apply course content directly to design and execute a tangible, locally-relevant plastic reduction initiative.</li> </ol> </div> == 📋 Project Information & Specifications == * '''School Affiliation:''' [[School:Environmental science|School of Environmental Science]] * '''Resource Classification:''' [[Wikiversity:Learning projects|Learning Project]] * '''Target Audience:''' Students, community activists, sustainability leaders, and concerned global citizens. No prior environmental science background required. * '''Format:''' 100% Asynchronous, open-access, self-paced OER learning. * '''Time Commitment:''' 3 to 4 hours per week for 5 weeks. * '''Required Tools:''' Notebook or digital log for plastic audits, camera/smartphone for field photos, and a free Wikiversity account. == 👥 Participant Roster & Cohort Sign-Up == Want to join this learning project? Add your Wikiversity username to the cohort list below! *(To sign up, edit this section and add <code># [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code> at the bottom)* # [[User:Earthinators|Earthinators Facilitator]] 14:26, 28 July 2026 (UTC) # ''(Sign up here to join the cohort!)'' == 🎯 Learning Outcomes == By the end of this course, participants will be able to: * '''Understand and Apply Systems Thinking:''' Use systems thinking to map the plastic lifecycle, global trade and waste flows, and environmental and health impacts. * '''Evaluate Governance Approaches:''' Discuss how feasible, effective, and fair different plastic policies and laws really are. * '''Launch a Local Action Project:''' Design and carry out a project targeting plastic pollution in your school, workplace, or community. * '''Reflect and Report:''' Analyze the results of your local project and write a 1 to 2 page reflection report sharing your outcomes and lessons learned. * '''Join a Global Community:''' Connect and collaborate with an international network of citizens taking action on plastic pollution. == 🗓️ 5-Week Action Journey & Comprehensive Syllabus == <div style="background: #f8fafc; border: 1px solid #cbd5e1; border-radius: 8px; padding: 12px 18px; margin-bottom: 20px;"> <b>🧭 Subpage Navigation:</b> [[/Week_1_Global_Plastic_System|Week 1: Global System]] • [[/Week_2_Ecological_and_Health_Impacts|Week 2: Impacts]] • [[/Week_3_Environmental_Policy_and_Governance|Week 3: Governance]] • [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives]] • [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Local Action]] </div> === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System - Oil Well to Household Waste]] === * '''Inquiry Question:''' Where does your plastic come from, and where does it go after disposal? * '''1.1 Petrochemical Origins and Industrial Expansion:''' Trace the transition from early synthetic resins like Parkesine and Bakelite to the post-World War II manufacturing boom that established modern mass disposable consumer culture. * '''1.2 Polymer Chemistry and Lifecycle Mapping:''' Examine how crude oil and natural gas refining feeds naphtha cracking into monomers, polymerization into resin pellets, and manufacturing into resin codes #1 PET through #7 OTHER. * '''1.3 Global Waste Leakage and Mass Balance Accounting:''' Quantify global municipal solid waste generation, mismanaged waste rates in developing infrastructure, and the breakdown of linear material throughput. * '''1.4 Environmental Justice and Global Waste Trade:''' Analyze environmental inequities in petrochemical refining corridors, international plastic waste dumping under the [[w:Basel Convention|Basel Convention]], and informal waste picker economies. * '''Weekly Milestone:''' Conduct a 24-hour personal plastic audit, log resin types (#1 to #7), and post your reflection on the <b>[[School talk:Environmental_science/Earthinators:_Plastic_Pollution_to_Circular_Solutions|Course Discussion Page]]</b> using <code>[[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 14:26, 28 July 2026 (UTC)</code>. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Ecological & Human Health Impacts]] === * '''Inquiry Question:''' How do plastics and microplastics move through food chains and natural ecosystems? * '''2.1 Climate Systems and Petrochemical Carbon Emissions:''' Calculate life-cycle greenhouse gas emissions across crude extraction, refining, high-heat incineration, and solar breakdown of marine plastics releasing methane. * '''2.2 Wildlife Ecotoxicology and Chemical Leaching:''' Investigate physical entanglement hazards, digestive blockages in marine species, and chemical leaching of additives like phthalates, [[w:Bisphenol A|BPA]], and [[w:Per- and polyfluoroalkyl substances|PFAS]] disrupting endocrine systems. * '''2.3 Microplastics and Human Health Hazards:''' Differentiate between primary microbeads, secondary synthetic textile fibers, bioaccumulation in marine food webs, and emerging microplastic presence in human blood and lung tissues. * '''2.4 Ocean Contamination and Hydrodynamics:''' Track river export vectors delivering land-based debris into oceanic gyres, forming accumulation zones like the [[w:Great Pacific garbage patch|Great Pacific Garbage Patch]] and deep-sea floor sinks. * '''Weekly Milestone:''' Observe a local park, storm drain, or body of water, map out potential plastic contamination pathways, and share your notes on the discussion page. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Environmental Policy & Corporate Governance]] === * '''Inquiry Question:''' Who is responsible for plastic waste, and what policies actually work? * '''3.1 Systems Dynamics and International Treaties:''' Map system feedback loops, identify high-leverage intervention points, and analyze structural progress under the [[w:UN Plastics Treaty|UN Global Plastics Treaty]]. * '''3.2 Extended Producer Responsibility (EPR) Frameworks:''' Evaluate polluter-pays legislative models that mandate manufacturer financial accountability for post-consumer waste management and packaging redesign. * '''3.3 Recycling Thermodynamics and Chemical Reprocessing Limits:''' Examine thermodynamic limits of mechanical recycling, polymer chain degradation, downcycling realities, and energy claims surrounding chemical pyrolysis. * '''3.4 Legislative Instruments and Container Deposit Systems:''' Measure the real-world impact of single-use plastic bag bans, retail fees, packaging recycled-content mandates, and [[w:Container-deposit legislation|Container Deposit Systems (DRS)]] achieving over 90 percent recovery. * '''Weekly Milestone:''' Post a 1-paragraph proposal for your final local community action project on the discussion page for peer feedback. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Materials Innovation & Greenwashing Precautions]] === * '''Inquiry Question:''' How can we distinguish genuine sustainable innovations from corporate greenwashing? * '''4.1 Upcycling Safety and Structural Repurposing:''' Evaluate structural upcycling techniques like ecobricking, structural plastic lumber, and chemical leaching risks when re-exposed to outdoor heat and UV radiation. * '''4.2 Environmental Art and Public Consciousness:''' Study how visual metrics, debris installations, and community art campaigns shift cultural perceptions and pressure policy makers. * '''4.3 Bioplastics, Bio-Based Polymers, and Degradation Limits:''' Analyze plant-derived polymers like [[w:Bioplastics|PLA and PHA]], distinguishing high-temperature industrial composting requirements (>58C) from natural environmental breakdown limits. * '''4.4 Deconstructing Greenwashing and Deceptive Eco-Claims:''' Audit corporate distraction campaigns, industry front groups, and [[w:Greenwashing|ISO 14021 environmental labeling standards]] to spot misleading eco-claims. * '''Weekly Milestone:''' Perform a quick greenwashing audit on an eco-labeled consumer product and post your critique on the discussion page. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Circular Economies & Local Community Action]] === * '''Inquiry Question:''' How do we transition from linear plastic waste to circular community solutions? * '''5.1 Zero-Waste Systems and Reuse Logistics:''' Explore neighborhood refill networks, repair cafés, package-free stores, and returnable container logistics. * '''5.2 Citizen Science and Corporate Brand Audits:''' Master empirical litter sampling protocols and [[w:Break Free From Plastic|Break Free From Plastic]] brand audit methodologies to trace corporate polluters. * '''5.3 Post-Linear Economics and Material Sufficiency:''' Apply industrial ecology concepts, [[w:Degrowth|degrowth economics]], and Product-as-a-Service models to reduce total material throughput. * '''5.4 Capstone Execution and Analytical Reflection:''' Finalize your community intervention, compile field data or photos, and write a 1 to 2 page reflection report analyzing project outcomes and systems leverage. * '''Final Milestone:''' Post your project summary, campaign materials, or reflection report on the course discussion page to complete the course. == 🏆 Assessment & Wikiversity Userbox Badge == Participants who complete all 5 weekly milestones and post their final 1 to 2 page reflection report on the course discussion page earn the '''Earthinators Circular Action Badge''' to display on their Wikiversity User page! <div style="background: #f1f5f9; border: 1px solid #cbd5e1; padding: 10px 14px; border-radius: 8px; width: fit-content;"> <code>{{User Wikiversity/Earthinators}}</code> — <b>Earthinators Circular Action Graduate</b> </div> == 📖 Open Educational Resources (OER) & Wikimedia Integration == === 🌿 Earthinators Initiative === * [[c:Category:Earthinators|Earthinators Media Gallery on Wikimedia Commons]] * [[d:Q103824308|Earthinators Knowledge Item on Wikidata]] === 🌊 Sister Wikimedia Projects & Portals === * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media & Videos on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] * [[b:Subject:Environmental sciences|Environmental Science Books on Wikibooks]] * [[q:Rachel Carson|Rachel Carson Quotes on Wikiquote]] <div style="border: 1px solid #cbd5e1; background: #f8fafc; padding: 12px 16px; border-radius: 8px; font-size: 0.88em; margin-top: 24px;"> <b>🔓 Open Educational Resource (OER) License:</b><br/> In full accordance with Wikiversity policy, all educational content on this page is licensed under the <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0)</a>. Feel free to reuse, adapt, and remix for non-commercial or commercial educational purposes. </div> [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] nw7xgpge8wzlb4b994ci8h306482snb 2819675 2819672 2026-07-28T15:13:46Z Earthinators 3102737 2819675 wikitext text/x-wiki {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] 9yjs5x31mnxot233zrfry0812j07urh 2819677 2819675 2026-07-28T15:16:15Z Atcovi 276019 Atcovi moved page [[School:Environmental science/Earthinators: Plastic Pollution to Circular Solutions]] to [[Earthinators: Plastic Pollution to Circular Solutions]] without leaving a redirect: reverting disruptive change 2819675 wikitext text/x-wiki {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] 9yjs5x31mnxot233zrfry0812j07urh 2819679 2819677 2026-07-28T15:17:52Z Atcovi 276019 project box(es) 2819679 wikitext text/x-wiki {{environmental science}} {{course}} {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] 1202hajz5yo2ybelyzkk4vwhqqqt6la 2819687 2819679 2026-07-28T15:46:38Z Earthinators 3102737 2819687 wikitext text/x-wiki {{Notice|1=<b>Seeking Guidance & Advice:</b> We are new to Wikiversity and building this open course. We warmly welcome advice, guidance, open resources, and feedback from experienced educators and community members on our [[Talk:Earthinators: Plastic Pollution to Circular Solutions|Discussion Page]].}} {{environmental science}} {{course}} {{Learning project}} {{Under construction}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] fxogsog21q12abprg2otwssk0ah0lbl 2819693 2819687 2026-07-28T17:15:06Z Earthinators 3102737 2819693 wikitext text/x-wiki {{Learning project}} {{course}} {{environmental science}} {{Under construction}} {{Notice|1=<b>Seeking Guidance & Peer Review:</b> We are new to Wikiversity and building this open course. We warmly welcome advice, guidance, open resources, and peer feedback from experienced educators and community members on our [[Talk:Earthinators: Plastic Pollution to Circular Solutions|Discussion Page]].}} {{Notice|1=<b>Thank You:</b> We express our sincere thanks and gratitude to Wikiversity educators and administrators for welcoming us and guiding our course development.}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] egadyo8fi2wra22k3y5l02vak8sreyr 2819694 2819693 2026-07-28T17:16:33Z Earthinators 3102737 2819694 wikitext text/x-wiki {{Learning project}} {{course}} {{environmental science}} {{OER}} {{Under construction}} {{Notice|1=<b>Seeking Guidance & Peer Review:</b> We are new to Wikiversity and building this open course. We warmly welcome advice, guidance, open resources, and peer feedback from experienced educators and community members on our [[Talk:Earthinators: Plastic Pollution to Circular Solutions|Discussion Page]].}} {{Notice|1=<b>Thank You:</b> We express our sincere thanks and gratitude to Wikiversity educators and administrators for welcoming us and guiding our course development.}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] szhhxsa5q124xo5tov53bsckbizdtl8 2819695 2819694 2026-07-28T17:17:53Z Earthinators 3102737 2819695 wikitext text/x-wiki {{OER}} {{Learning project}} {{course}} {{environmental science}} {{Under construction}} {{Notice|1=<b>Seeking Guidance & Peer Review:</b> We are new to Wikiversity and building this open course. We warmly welcome advice, guidance, open resources, and peer feedback from experienced educators and community members on our [[Talk:Earthinators: Plastic Pollution to Circular Solutions|Discussion Page]].}} {{Notice|1=<b>Thank You:</b> We express our sincere thanks and gratitude to Wikiversity educators and administrators for welcoming us and guiding our course development.}} = Earthinators: Plastic Systems, Governance & Circular Solutions = ''A 5-week project on plastic lifecycles, global impacts, policies, and community action.'' [[File:Cow eating plastic waste beside roadside vegetation.jpg|thumb|300px|Cow eating plastic waste along a roadside.]] == About the Course == Plastics are more than just waste. They help us understand how global environmental, social, and economic systems work. '''Earthinators: Plastic Systems, Governance & Circular Solutions''' is a free 5-week learning project. You will trace plastic from oil wells to oceans, policy frameworks, and local solutions. You will also launch a project in your school, workplace, or community. This could mean advocating for a plastic bag ban, starting a refill campaign, or organizing a brand audit. This course is completely open, self-paced, and collaborative. == Core Educational Principles == # '''Learning is Social:''' Participants learn by sharing ideas and giving feedback on discussion pages. # '''Learning Leads to Action:''' Every lesson helps you launch a real project in your community. == Project Details == * '''School:''' [[School:Environmental science|School of Environmental Science]] * '''Type:''' [[Wikiversity:Learning projects|Learning Project]] * '''Time:''' 3 to 4 hours per week for 5 weeks. * '''Format:''' Self-paced, asynchronous learning. == Participants == Add your signature below to join the cohort: # [[User:Earthinators|Earthinators Facilitator]] 15:13, 28 July 2026 (UTC) == Learning Outcomes == By the end of this course, you will be able to: * '''Apply Systems Thinking:''' Map the plastic lifecycle, global waste flows, and environmental impacts. * '''Evaluate Policies:''' Discuss real laws like Extended Producer Responsibility, plastic bag bans, and bottle deposit systems. * '''Launch a Project:''' Carry out a plastic reduction project in your local area. * '''Reflect & Share:''' Write a 1 to 2 page report sharing your results and lessons learned. * '''Connect Globally:''' Work with an international group taking action against plastic pollution. == 5-Week Outline == === [[/Week_1_Global_Plastic_System|Week 1: The Global Plastic System]] === * '''1.1 History of Plastics:''' From Bakelite to modern throwaway culture. * '''1.2 Plastic Lifecycle:''' From oil refining to resin codes (#1 PET to #7 OTHER). * '''1.3 Global Waste Flows:''' How mismanaged trash enters rivers and oceans. * '''1.4 Environmental Justice:''' Refining impacts, waste trade, and waste pickers. === [[/Week_2_Ecological_and_Health_Impacts|Week 2: Environmental & Health Impacts]] === * '''2.1 Climate Connections:''' Carbon emissions from production and trash burning. * '''2.2 Ecosystem Hazards:''' Wildlife entanglement, ingestion, and chemical leaching. * '''2.3 Microplastics Crisis:''' Microbeads, synthetic fibers, and human exposure. * '''2.4 Ocean Pollution:''' River vectors and ocean garbage patches. === [[/Week_3_Environmental_Policy_and_Governance|Week 3: Plastic Policy & Governance]] === * '''3.1 Systems Thinking:''' Leverage points and the UN Global Plastics Treaty. * '''3.2 Extended Producer Responsibility (EPR):''' Making manufacturers pay for waste. * '''3.3 Recycling Limits:''' Mechanical recycling constraints and chemical recycling claims. * '''3.4 Real-World Laws:''' Evaluating bag bans, fees, and deposit systems. === [[/Week_4_Materials_Innovation_and_Greenwashing|Week 4: Alternatives & Greenwashing]] === * '''4.1 Upcycling:''' Ecobricks, structural reuse, and chemical safety. * '''4.2 Environmental Art:''' Using art installations for community awareness. * '''4.3 Bioplastics:''' Plant plastics (PLA, PHA) and industrial composting needs. * '''4.4 Greenwashing:''' Spotting fake eco-claims and reading labels. === [[/Week_5_Circular_Economies_and_Local_Action|Week 5: Taking Local Action]] === * '''5.1 Zero-Waste Models:''' Refill shops, repair cafés, and reusable container systems. * '''5.2 Citizen Science:''' Cleanups and corporate brand audits. * '''5.3 Beyond Plastic:''' Degrowth, redesign, and material reduction. * '''5.4 Capstone Project:''' Launching your project and sharing your reflection report. == Related Resources == * [[w:Plastic pollution|Plastic Pollution Article on Wikipedia]] * [[c:Category:Plastic_pollution|Plastic Pollution Media on Wikimedia Commons]] * [[d:Q7202090|Plastic Pollution Data on Wikidata]] * [[w:Circular economy|Circular Economy Portal on Wikipedia]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] 4b7azhoktj2j2wxqs84eikdbd4ku7qw 3 330711 2819644 2819640 2026-07-28T12:30:09Z Solstag 64708 /* Welcome! */ Reply 2819644 wikitext text/x-wiki == Welcome! == Ni! Welcome @[[User:Dnshitobu|Dnshitobu]], let me know if I can lend you a hand. [[User:Solstag|Solstag]] ([[User talk:Solstag|discuss]] • [[Special:Contributions/Solstag|contribs]]) 00:25, 24 July 2026 (UTC) :@[[User:Solstag|Solstag]] :Thank you so much for taking me through it. I look forward to expanding the content here. Please, you can keep an eye on my content and suggest best practices [[User:Dnshitobu|Dnshitobu]] ([[User talk:Dnshitobu|discuss]] • [[Special:Contributions/Dnshitobu|contribs]]) 10:44, 28 July 2026 (UTC) ::Right. You're welcome! I have a few observations given how you're deploying your course. I'll leave them in the talk page for the course. [[User:Solstag|Solstag]] ([[User talk:Solstag|discuss]] • [[Special:Contributions/Solstag|contribs]]) 12:30, 28 July 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Dnshitobu!'''|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]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:13, 27 July 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} nmeq1syg4nzhjswarbltq0g4hzet4ol 0 330751 2819643 2819429 2026-07-28T12:18:42Z Solstag 64708 /* Practical Projects & Assessment Guidelines */ 2819643 wikitext text/x-wiki __TOC__ = Computing for Senior High School Year 2 = Welcome to the '''Year 2 Computing Portal''' for Senior High Schools in Ghana. This page provides the curriculum roadmap, weekly topic breakdown, and learning resources designed in accordance with the National Council for Curriculum and Assessment (NaCCA) Standards-Based Curriculum. == Course Overview == The SHS Year 2 Computing course builds upon Year 1 fundamentals by deepening students' technical competency in digital logic, network engineering, algorithm design, real-world programming, machine learning basics, and relational database systems. Key focus areas for this year include: * '''Boolean Algebra & Digital Logic:''' Designing and analyzing logic circuits, evaluating Boolean expressions, and understanding intermediate gate behaviors. * '''Hardware, Software & Scanning Technologies:''' Exploring software categories alongside modern data encoding using Barcodes and QR codes. * '''Network Systems & Connectivity:''' In-depth examination of network architectures, physical connection media, and network topology implementations. * '''Data Structures & Memory Operations:''' Practical mapping of arrays, stacks, and queues directly to main memory (RAM) management. * '''Applied Programming & Machine Learning:''' Developing practical software applications in Python, building utility tools (e.g., custom calculators), and introducing basic machine learning logic and ethics. * '''Web Page Editors & Database Design:''' Constructing web pages using modern web editors and structuring relational databases. == Weekly Syllabus Schedule == {| class="wikitable sortable" |- ! Week !! Section !! Focal Area / Topic |- | Week 1 || Section 1: Boolean Algebra || [[/Understanding Logic Operations and Expressions|Understanding Logic Operations and Expressions]] |- | Week 2 || Section 1: Boolean Algebra || [[/Applying Boolean Expressions|Applying Boolean Expressions]] |- | Week 3 || Section 1: Boolean Algebra || [[/Arithmetic Operations and Logic Circuit Construction|Arithmetic Operations and Logic Circuit Construction]] |- | Week 4 || Section 2: Hardware, Software & Barcodes || [[/Creating and Utilising Barcodes and QR Codes|Creating and Utilising Barcodes and QR Codes]] |- | Week 5 || Section 2: Hardware, Software & Barcodes || [[/Barcodes and QR Codes Applications|Barcodes and QR Codes Applications]] |- | Week 6 || Section 2: Hardware, Software & Barcodes || [[/Types of Software and Their Functions|Types of Software and Their Functions]] |- | Week 7 || Section 3: Network Systems & Topologies || [[/Network Systems Architecture|Network Systems Architecture]] |- | Week 8 || Section 3: Network Systems & Topologies || [[/Network Topologies and Configurations|Network Topologies and Configurations]] |- | Week 9 || Section 3: Network Systems & Topologies || [[/Network Connectivity and Media|Network Connectivity and Media]] |- | Week 10 || Section 3: Network Systems & Topologies || [[/Network System Connectivity and Diagnostics|Network System Connectivity and Diagnostics]] |- | Week 11 || Section 3: Network Systems & Topologies || [[/Advanced Data Communication Networks|Advanced Data Communication Networks]] |- | Week 12 || Section 4: Main Memory & Data Structures || [[/Main Memory and Data Structures|Main Memory and Data Structures]] |- | Week 13 || Section 4: Main Memory & Data Structures || [[/Real-Life Applications of Data Structures|Real-Life Applications of Data Structures]] |- | Week 14 || Section 5: Utilising Programming Language || [[/Basic Programming Concepts|Basic Programming Concepts]] |- | Week 15 || Section 5: Utilising Programming Language || [[/Control Structures and Program Logic|Control Structures and Program Logic]] |- | Week 16 || Section 5: Utilising Programming Language || [[/Creating Practical Applications (Simple Calculator)|Creating Practical Applications (Simple Calculator)]] |- | Week 17 || Section 5: Utilising Programming Language || [[/Understanding Machine Learning Concepts|Understanding Machine Learning Concepts]] |- | Week 18 || Section 5: Utilising Programming Language || [[/Implementing Machine Learning and Ethical Considerations|Implementing Machine Learning and Ethical Considerations]] |- | Week 19 || Section 6: Web Page Editors & Databases || [[/Page Editors and Design Elements|Page Editors and Design Elements]] |- | Week 20 || Section 6: Web Page Editors & Databases || [[/Responsive Web Layouts and Navigation|Responsive Web Layouts and Navigation]] |- | Week 21 || Section 6: Web Page Editors & Databases || [[/Web Publishing and Asset Management|Web Publishing and Asset Management]] |- | Week 22 || Section 6: Web Page Editors & Databases || [[/Introduction to Database Concepts|Introduction to Database Concepts]] |- | Week 23 || Section 6: Web Page Editors & Databases || [[/Data Modeling and Entity Relationships|Data Modeling and Entity Relationships]] |- | Week 24 || Section 6: Web Page Editors & Databases || [[/Relational Database Design|Relational Database Design]] |} == Practical Projects & Assessment Guidelines == According to the NaCCA assessment framework, Year 2 students complete both individual laboratory tasks and collaborative group projects throughout the academic year: 1. '''Logic Circuit Construction Project:''' Design, build, and test a multi-input logic circuit combining NAND, NOR, and XOR gates for real-world automated input systems. 2. '''Software Application Project:''' Develop a functional GUI or command-line utility (e.g., a multi-function calculator) using Python control structures. 3. '''Database & Web Layout Design:''' Model an entity-relationship (ER) diagram for a school record system and design a corresponding interactive web view using visual web tools. == References & Official Resources == * '''Ministry of Education / NaCCA (2023).''' ''Computing Curriculum for Secondary Education (SHS 1 – 3)''. Accra: National Council for Curriculum and Assessment. * '''Ministry of Education / T-TEL (2024).''' ''Year Two Computing Teacher Manual''. Ghana Education Service Resource Packs. * '''National Pre-Tertiary Learning Assessment Framework (NPLAF).''' NaCCA Curriculum Portal ([https://nacca.gov.gh nacca.gov.gh]). [[Category:Computing]] [[Category:Ghana]] [[Category:Ghana Education Service]] [[Category:Secondary Education]] gnhyrfvk6xaz4uypms145xh1g9wdfg5 0 330755 2819682 2819525 2026-07-28T15:32:51Z DavidMCEddy 218607 /* Bibliography */ add another book chaper 2819682 wikitext text/x-wiki :''This discusses a 2026-07-31 interview with Yonty Friesem<ref name=Yonty><!--Yonty Friesem-->{{cite Q|Q140685186}}</ref> about media education, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-08 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Yonty Friesem on media education.webm|thumb|2026-07-31 interview with Yonty Friesem about media education.]]--> <!--[[File:Yonty Friesem on media education.ogg|thumb|29:00 mm:ss excerpted from a 2026-07-31 interview with Yonty Friesem about media education.]]--> Yonty Friesem<ref name=Yonty/> discusses media literacy education and the Media Education Lab,<ref name=MEL><!--Media Education Lab-->{{cite Q|Q140686986}}</ref> of which he is Executive Director. Friesem is an award-winning educator, scholar and filmmaker and an Associate Professor of Communication at [[w:Columbia College Chicago|Columbia College Chicago]]. The Media Education Lab is an online community working to advance digital and media literacy education through leadership development, scholarship, and community engagement. It was founded in 2003 by by [[w:Renee Hobbs|Renee Hobbs]], then at [[w:Temple University|Temple University]]. It has team members, faculty and graduate students in many different countries. Friesem completed a 4-year program in Video and Film Production at the Camera Obscura School of Art in Israel in 2001, a BA in Social Science and Humanities at the [[w:Open University of Israel|Open University of Israel]] in 2007, an MA in Management and Leadership in Education at [[w:Tel Aviv University|Tel Aviv University]] in 2010, and a PhD in Education at the [[w:University of Rhode Island|University of Rhode Island]] in 2015. (Hobbs was appointed the Founding Director of the Harrington School of Communication and Media at the University of Rhode Island in 2012.) Friesem joined Columbia College Chicago in 2018. He is lead editor of (2022) ''The Routledge handbook on media education futures post-pandemic''. And he is author or a co-author of numerous other publications including the following: * (2026) "Young people evaluating online information credibility. Special issue synthesis"<ref>Nestor et al. (2026).</ref> * (2023) "Social Justice Education as an Intercultural Experience for Foreign Students in the United States"<ref>Yeh and Friesem (2023).</ref> * (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication"<ref>Friesem and Friesem (2021).</ref> * (2021-07-14) "The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences"<ref>Y. Friesem (2021).</ref> * (2020-10-13) "It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning"<ref>Friesem (2020-10-13).</ref> * (2020-08-04) "Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media"<ref>Friesem and Greene (2020).</ref> Friesem is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Yonty Friesem, ed. (2022) The Routledge handbook on media education futures post-pandemic-->{{cite Q|Q140705564|author=Yonty Friesem, ed.}} * <!--Yonty Friesem (2021-07-14) The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences-->{{cite Q|Q140706700|chapter=ch. 3 in Fisher, ed.}} * <!--Yonty Friesem (2020-10-13) It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning-->{{cite Q|Q140707050}} * <!--Elizaveta Friesem and Yonty Friesem (2021-01-01) Media Literacy Education in the Era of Post-Truth: Paradigm Crisis-->{{cite Q|Q140738490|chapter=ch. 33 in Khosrow-pour et al., eds.}} * <!--Yonty Friesem and Elizaveta Friesem (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication-->{{cite Q|Q140706479}} * <!--Yonty Friesem and Kelsey Greene (2020-08-04) Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media-->{{cite Q|Q140707249}} * <!-- Nicolae Nistor, Yonty Friesem, Cristina Nistor, and Rares Beuran (2026-06-18) "Young people evaluating online information credibility. Special issue synthesis", Computers in Human Behavior-->{{cite Q|Q140705851}} * <!--Ellen Yeh and Yonty Friesem (2023) Social Justice Education as an Intercultural Experience for Foreign Students in the United States-->{{cite Q|Q129327420}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Education]] [[Category:Media literacy]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> rpjl8dxvr0d6clauiryd63fiz8i0y3m 2819683 2819682 2026-07-28T15:36:51Z DavidMCEddy 218607 /* Bibliography */ add ref bk 2819683 wikitext text/x-wiki :''This discusses a 2026-07-31 interview with Yonty Friesem<ref name=Yonty><!--Yonty Friesem-->{{cite Q|Q140685186}}</ref> about media education, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-08 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Yonty Friesem on media education.webm|thumb|2026-07-31 interview with Yonty Friesem about media education.]]--> <!--[[File:Yonty Friesem on media education.ogg|thumb|29:00 mm:ss excerpted from a 2026-07-31 interview with Yonty Friesem about media education.]]--> Yonty Friesem<ref name=Yonty/> discusses media literacy education and the Media Education Lab,<ref name=MEL><!--Media Education Lab-->{{cite Q|Q140686986}}</ref> of which he is Executive Director. Friesem is an award-winning educator, scholar and filmmaker and an Associate Professor of Communication at [[w:Columbia College Chicago|Columbia College Chicago]]. The Media Education Lab is an online community working to advance digital and media literacy education through leadership development, scholarship, and community engagement. It was founded in 2003 by by [[w:Renee Hobbs|Renee Hobbs]], then at [[w:Temple University|Temple University]]. It has team members, faculty and graduate students in many different countries. Friesem completed a 4-year program in Video and Film Production at the Camera Obscura School of Art in Israel in 2001, a BA in Social Science and Humanities at the [[w:Open University of Israel|Open University of Israel]] in 2007, an MA in Management and Leadership in Education at [[w:Tel Aviv University|Tel Aviv University]] in 2010, and a PhD in Education at the [[w:University of Rhode Island|University of Rhode Island]] in 2015. (Hobbs was appointed the Founding Director of the Harrington School of Communication and Media at the University of Rhode Island in 2012.) Friesem joined Columbia College Chicago in 2018. He is lead editor of (2022) ''The Routledge handbook on media education futures post-pandemic''. And he is author or a co-author of numerous other publications including the following: * (2026) "Young people evaluating online information credibility. Special issue synthesis"<ref>Nestor et al. (2026).</ref> * (2023) "Social Justice Education as an Intercultural Experience for Foreign Students in the United States"<ref>Yeh and Friesem (2023).</ref> * (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication"<ref>Friesem and Friesem (2021).</ref> * (2021-07-14) "The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences"<ref>Y. Friesem (2021).</ref> * (2020-10-13) "It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning"<ref>Friesem (2020-10-13).</ref> * (2020-08-04) "Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media"<ref>Friesem and Greene (2020).</ref> Friesem is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Yonty Friesem, ed. (2022) The Routledge handbook on media education futures post-pandemic-->{{cite Q|Q140705564|author=Yonty Friesem, ed.}} * <!--Yonty Friesem (2021-07-14) The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences-->{{cite Q|Q140706700|chapter=ch. 3 in Fisher, ed.}} * <!--Yonty Friesem (2020-10-13) It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning-->{{cite Q|Q140707050}} * <!--Elizaveta Friesem and Yonty Friesem (2021-01-01) Media Literacy Education in the Era of Post-Truth: Paradigm Crisis-->{{cite Q|Q140738490|chapter=ch. 33 in Khosrow-pour et al., eds.}} * <!--Yonty Friesem and Elizaveta Friesem (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication-->{{cite Q|Q140706479}} * <!--Yonty Friesem and Kelsey Greene (2020-08-04) Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media-->{{cite Q|Q140707249}} * <!--Mehdi Khosrowpour et al., eds. (Research Anthology on Fake News, Political Warfare, and Combatting the Spread of Misinformation-->{{cite Q|Q140738510|author=Mehdi Khosrowpour et al., eds.}} * <!-- Nicolae Nistor, Yonty Friesem, Cristina Nistor, and Rares Beuran (2026-06-18) "Young people evaluating online information credibility. Special issue synthesis", Computers in Human Behavior-->{{cite Q|Q140705851}} * <!--Ellen Yeh and Yonty Friesem (2023) Social Justice Education as an Intercultural Experience for Foreign Students in the United States-->{{cite Q|Q129327420}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Education]] [[Category:Media literacy]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> 93wv9bfegeya2f9pd7rjlur40hafikp 2819684 2819683 2026-07-28T15:40:27Z DavidMCEddy 218607 /* Bibliography */ add Fisher, ed 2819684 wikitext text/x-wiki :''This discusses a 2026-07-31 interview with Yonty Friesem<ref name=Yonty><!--Yonty Friesem-->{{cite Q|Q140685186}}</ref> about media education, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-08 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Yonty Friesem on media education.webm|thumb|2026-07-31 interview with Yonty Friesem about media education.]]--> <!--[[File:Yonty Friesem on media education.ogg|thumb|29:00 mm:ss excerpted from a 2026-07-31 interview with Yonty Friesem about media education.]]--> Yonty Friesem<ref name=Yonty/> discusses media literacy education and the Media Education Lab,<ref name=MEL><!--Media Education Lab-->{{cite Q|Q140686986}}</ref> of which he is Executive Director. Friesem is an award-winning educator, scholar and filmmaker and an Associate Professor of Communication at [[w:Columbia College Chicago|Columbia College Chicago]]. The Media Education Lab is an online community working to advance digital and media literacy education through leadership development, scholarship, and community engagement. It was founded in 2003 by by [[w:Renee Hobbs|Renee Hobbs]], then at [[w:Temple University|Temple University]]. It has team members, faculty and graduate students in many different countries. Friesem completed a 4-year program in Video and Film Production at the Camera Obscura School of Art in Israel in 2001, a BA in Social Science and Humanities at the [[w:Open University of Israel|Open University of Israel]] in 2007, an MA in Management and Leadership in Education at [[w:Tel Aviv University|Tel Aviv University]] in 2010, and a PhD in Education at the [[w:University of Rhode Island|University of Rhode Island]] in 2015. (Hobbs was appointed the Founding Director of the Harrington School of Communication and Media at the University of Rhode Island in 2012.) Friesem joined Columbia College Chicago in 2018. He is lead editor of (2022) ''The Routledge handbook on media education futures post-pandemic''. And he is author or a co-author of numerous other publications including the following: * (2026) "Young people evaluating online information credibility. Special issue synthesis"<ref>Nestor et al. (2026).</ref> * (2023) "Social Justice Education as an Intercultural Experience for Foreign Students in the United States"<ref>Yeh and Friesem (2023).</ref> * (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication"<ref>Friesem and Friesem (2021).</ref> * (2021-07-14) "The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences"<ref>Y. Friesem (2021).</ref> * (2020-10-13) "It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning"<ref>Friesem (2020-10-13).</ref> * (2020-08-04) "Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media"<ref>Friesem and Greene (2020).</ref> Friesem is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Joshua A. Fisher, ed. (2021) Augmented and Mixed Reality for Communities-->{{cite Q|Q140706704|author=Joshua A. Fisher, ed.}} * <!--Yonty Friesem, ed. (2022) The Routledge handbook on media education futures post-pandemic-->{{cite Q|Q140705564|author=Yonty Friesem, ed.}} * <!--Yonty Friesem (2021-07-14) The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences-->{{cite Q|Q140706700|chapter=ch. 3 in Fisher, ed.}} * <!--Yonty Friesem (2020-10-13) It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning-->{{cite Q|Q140707050}} * <!--Elizaveta Friesem and Yonty Friesem (2021-01-01) Media Literacy Education in the Era of Post-Truth: Paradigm Crisis-->{{cite Q|Q140738490|chapter=ch. 33 in Khosrow-pour et al., eds.}} * <!--Yonty Friesem and Elizaveta Friesem (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication-->{{cite Q|Q140706479}} * <!--Yonty Friesem and Kelsey Greene (2020-08-04) Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media-->{{cite Q|Q140707249}} * <!--Mehdi Khosrowpour et al., eds. (Research Anthology on Fake News, Political Warfare, and Combatting the Spread of Misinformation-->{{cite Q|Q140738510|author=Mehdi Khosrowpour et al., eds.}} * <!-- Nicolae Nistor, Yonty Friesem, Cristina Nistor, and Rares Beuran (2026-06-18) "Young people evaluating online information credibility. Special issue synthesis", Computers in Human Behavior-->{{cite Q|Q140705851}} * <!--Ellen Yeh and Yonty Friesem (2023) Social Justice Education as an Intercultural Experience for Foreign Students in the United States-->{{cite Q|Q129327420}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Education]] [[Category:Media literacy]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> d3695sw50uxe9ckjooxq5bzhc69gurj 2819685 2819684 2026-07-28T15:41:08Z DavidMCEddy 218607 /* Bibliography */ sort bib 2819685 wikitext text/x-wiki :''This discusses a 2026-07-31 interview with Yonty Friesem<ref name=Yonty><!--Yonty Friesem-->{{cite Q|Q140685186}}</ref> about media education, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-08 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Yonty Friesem on media education.webm|thumb|2026-07-31 interview with Yonty Friesem about media education.]]--> <!--[[File:Yonty Friesem on media education.ogg|thumb|29:00 mm:ss excerpted from a 2026-07-31 interview with Yonty Friesem about media education.]]--> Yonty Friesem<ref name=Yonty/> discusses media literacy education and the Media Education Lab,<ref name=MEL><!--Media Education Lab-->{{cite Q|Q140686986}}</ref> of which he is Executive Director. Friesem is an award-winning educator, scholar and filmmaker and an Associate Professor of Communication at [[w:Columbia College Chicago|Columbia College Chicago]]. The Media Education Lab is an online community working to advance digital and media literacy education through leadership development, scholarship, and community engagement. It was founded in 2003 by by [[w:Renee Hobbs|Renee Hobbs]], then at [[w:Temple University|Temple University]]. It has team members, faculty and graduate students in many different countries. Friesem completed a 4-year program in Video and Film Production at the Camera Obscura School of Art in Israel in 2001, a BA in Social Science and Humanities at the [[w:Open University of Israel|Open University of Israel]] in 2007, an MA in Management and Leadership in Education at [[w:Tel Aviv University|Tel Aviv University]] in 2010, and a PhD in Education at the [[w:University of Rhode Island|University of Rhode Island]] in 2015. (Hobbs was appointed the Founding Director of the Harrington School of Communication and Media at the University of Rhode Island in 2012.) Friesem joined Columbia College Chicago in 2018. He is lead editor of (2022) ''The Routledge handbook on media education futures post-pandemic''. And he is author or a co-author of numerous other publications including the following: * (2026) "Young people evaluating online information credibility. Special issue synthesis"<ref>Nestor et al. (2026).</ref> * (2023) "Social Justice Education as an Intercultural Experience for Foreign Students in the United States"<ref>Yeh and Friesem (2023).</ref> * (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication"<ref>Friesem and Friesem (2021).</ref> * (2021-07-14) "The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences"<ref>Y. Friesem (2021).</ref> * (2020-10-13) "It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning"<ref>Friesem (2020-10-13).</ref> * (2020-08-04) "Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media"<ref>Friesem and Greene (2020).</ref> Friesem is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Joshua A. Fisher, ed. (2021) Augmented and Mixed Reality for Communities-->{{cite Q|Q140706704|author=Joshua A. Fisher, ed.}} * <!--Elizaveta Friesem and Yonty Friesem (2021-01-01) Media Literacy Education in the Era of Post-Truth: Paradigm Crisis-->{{cite Q|Q140738490|chapter=ch. 33 in Khosrow-pour et al., eds.}} * <!--Yonty Friesem, ed. (2022) The Routledge handbook on media education futures post-pandemic-->{{cite Q|Q140705564|author=Yonty Friesem, ed.}} * <!--Yonty Friesem (2021-07-14) The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences-->{{cite Q|Q140706700|chapter=ch. 3 in Fisher, ed.}} * <!--Yonty Friesem (2020-10-13) It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning-->{{cite Q|Q140707050}} * <!--Yonty Friesem and Elizaveta Friesem (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication-->{{cite Q|Q140706479}} * <!--Yonty Friesem and Kelsey Greene (2020-08-04) Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media-->{{cite Q|Q140707249}} * <!--Mehdi Khosrowpour et al., eds. (Research Anthology on Fake News, Political Warfare, and Combatting the Spread of Misinformation-->{{cite Q|Q140738510|author=Mehdi Khosrowpour et al., eds.}} * <!-- Nicolae Nistor, Yonty Friesem, Cristina Nistor, and Rares Beuran (2026-06-18) "Young people evaluating online information credibility. Special issue synthesis", Computers in Human Behavior-->{{cite Q|Q140705851}} * <!--Ellen Yeh and Yonty Friesem (2023) Social Justice Education as an Intercultural Experience for Foreign Students in the United States-->{{cite Q|Q129327420}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Education]] [[Category:Media literacy]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> irqhebt8w92i63atqlshlipa15wyrtj 2819688 2819685 2026-07-28T15:46:46Z DavidMCEddy 218607 add Y. Friesem and E. Friesem 2819688 wikitext text/x-wiki :''This discusses a 2026-07-31 interview with Yonty Friesem<ref name=Yonty><!--Yonty Friesem-->{{cite Q|Q140685186}}</ref> about media education, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-08-08 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Yonty Friesem on media education.webm|thumb|2026-07-31 interview with Yonty Friesem about media education.]]--> <!--[[File:Yonty Friesem on media education.ogg|thumb|29:00 mm:ss excerpted from a 2026-07-31 interview with Yonty Friesem about media education.]]--> Yonty Friesem<ref name=Yonty/> discusses media literacy education and the Media Education Lab,<ref name=MEL><!--Media Education Lab-->{{cite Q|Q140686986}}</ref> of which he is Executive Director. Friesem is an award-winning educator, scholar and filmmaker and an Associate Professor of Communication at [[w:Columbia College Chicago|Columbia College Chicago]]. The Media Education Lab is an online community working to advance digital and media literacy education through leadership development, scholarship, and community engagement. It was founded in 2003 by by [[w:Renee Hobbs|Renee Hobbs]], then at [[w:Temple University|Temple University]]. It has team members, faculty and graduate students in many different countries. Friesem completed a 4-year program in Video and Film Production at the Camera Obscura School of Art in Israel in 2001, a BA in Social Science and Humanities at the [[w:Open University of Israel|Open University of Israel]] in 2007, an MA in Management and Leadership in Education at [[w:Tel Aviv University|Tel Aviv University]] in 2010, and a PhD in Education at the [[w:University of Rhode Island|University of Rhode Island]] in 2015. (Hobbs was appointed the Founding Director of the Harrington School of Communication and Media at the University of Rhode Island in 2012.) Friesem joined Columbia College Chicago in 2018. He is lead editor of (2022) ''The Routledge handbook on media education futures post-pandemic''. And he is author or a co-author of numerous other publications including the following: * (2026) "Young people evaluating online information credibility. Special issue synthesis"<ref>Nestor et al. (2026).</ref> * (2023) "Social Justice Education as an Intercultural Experience for Foreign Students in the United States"<ref>Yeh and Friesem (2023).</ref> * (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication"<ref>Y. Friesem and E. Friesem (2021).</ref> * (2021-07-14) "The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences"<ref>Y. Friesem (2021).</ref> * (2021) "Media Literacy Education in the Era of Post-Truth: Paradigm Crisis"<ref>E. Friesem and Y. Friesem (2021).</ref> * (2020-10-13) "It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning"<ref>Friesem (2020-10-13).</ref> * (2020-08-04) "Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media"<ref>Friesem and Greene (2020).</ref> Friesem is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Joshua A. Fisher, ed. (2021) Augmented and Mixed Reality for Communities-->{{cite Q|Q140706704|author=Joshua A. Fisher, ed.}} * <!--Elizaveta Friesem and Yonty Friesem (2021-01-01) Media Literacy Education in the Era of Post-Truth: Paradigm Crisis-->{{cite Q|Q140738490|chapter=ch. 33 in Khosrow-pour et al., eds.}} * <!--Yonty Friesem, ed. (2022) The Routledge handbook on media education futures post-pandemic-->{{cite Q|Q140705564|author=Yonty Friesem, ed.}} * <!--Yonty Friesem (2021-07-14) The PARIS model: Creating a sustainable and participatory civic media with and for the community through immersive experiences-->{{cite Q|Q140706700|chapter=ch. 3 in Fisher, ed.}} * <!--Yonty Friesem (2020-10-13) It’s All About Control: How Giving Kids Control Over Access, Content, and Format of Their Media Production Advances Social and Emotional Learning-->{{cite Q|Q140707050}} * <!--Yonty Friesem and Elizaveta Friesem (2021-12-29) "The secret sauce of online community of practice during COVID-19 pandemic: Nonviolent Communication-->{{cite Q|Q140706479}} * <!--Yonty Friesem and Kelsey Greene (2020-08-04) Tuned In: The Importance of Peer Feedback with Foster Youth Creating Media-->{{cite Q|Q140707249}} * <!--Mehdi Khosrowpour et al., eds. (Research Anthology on Fake News, Political Warfare, and Combatting the Spread of Misinformation-->{{cite Q|Q140738510|author=Mehdi Khosrowpour et al., eds.}} * <!-- Nicolae Nistor, Yonty Friesem, Cristina Nistor, and Rares Beuran (2026-06-18) "Young people evaluating online information credibility. Special issue synthesis", Computers in Human Behavior-->{{cite Q|Q140705851}} * <!--Ellen Yeh and Yonty Friesem (2023) Social Justice Education as an Intercultural Experience for Foreign Students in the United States-->{{cite Q|Q129327420}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Education]] [[Category:Media literacy]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> b9gxedkelhvxvms0kgh20ubpqli8yg8 0 330774 2819745 2819550 2026-07-29T04:02:29Z JackBot 238563 Bot: Fixing double redirect from [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Fracking: Tracing Climate Change Disinformation on Social Media]] to [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Fracking: tracing climate change disinformation on social media]] 2819745 wikitext text/x-wiki #REDIRECT [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Fracking: tracing climate change disinformation on social media]] hy9kgnrhu37ce5rwpm5okgob168o9dz 6 330797 2819646 2026-07-28T12:40:09Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819646 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |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}} 6xe0jdo2p4ypkwm771jfad7yx2fdz4b 1 330798 2819647 2026-07-28T13:06:42Z Solstag 64708 /* Subpage names */ new section 2819647 wikitext text/x-wiki == Subpage names == Hi @[[User:Dnshitobu|Dnshitobu]], I leave my observations here. === First observation: courses === We had named the subpages "…/one" and "…/two", but seeing the contents of the main page now I think we could rename them "…/Year one" and "…/Year two". And in my view it would be even better if you give them names specific to the year you'll be teaching, so "…/Year one/Class of 2026". Where you'd have the program as you're teaching it in 2026. Because in future years it can evolve, and next year you'd start "…/Year one/Class of 2027", maybe by copying the contents of 2026 or maybe by preparing something new, which may be the case if another teacher comes in and prefers to organize the course differently. And this way you keep the courses for the past classes available. You can move the pages in the Tools menu, but let me know if there's any trouble I can move them for you. === Second observation: lessons === The pages for the lessons are currently named like "…/Year one/''Lesson name''". But in the future a lesson can move from one level to another, for several reasons. Moreover, even in the same level they can be specific to one class you're teaching, and you'd it slightly different in another. So you might want to keep each lesson for each class as a separate page, by naming them like this: "…/Year one/Class of 2026/''Lesson name''". Then each class would have it's own copy of the lesson, which is also a good thing if you want them to have a clear discussion page in which to leave questions about the lesson. But, if you don't want one copy of the lesson for each class, then for a “generic lesson” I suggest you avoid binding levels and lessons in the name. In this case you could rename the lessons as "…/lessons/''Lesson name''" and just link them in the page of courses that will use them. (And you could still have a clear discussion page if you create "…/Year one/Class of 2026/''Lesson name''" as a [[mw:Transclusion|transclusion]] of "…/lessons/''Lesson name''".) I hope this is clear! Let me know if you have questions. Cheers, [[User:Solstag|Solstag]] ([[User talk:Solstag|discuss]] • [[Special:Contributions/Solstag|contribs]]) 13:06, 28 July 2026 (UTC) 75c1uebeaeync0l2y6leb7ip9khzcl3 6 330799 2819652 2026-07-28T14:01:16Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B Single Level (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819652 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B Single Level (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |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}} kg6n2gncq4oqwkv4bpvf7a05f20zhw2 6 330800 2819653 2026-07-28T14:02:10Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2C Multi-Level (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819653 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2C Multi-Level (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |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}} nyjfcwjw32sj4ytx4oygq9ohvqmlt4k 6 330801 2819655 2026-07-28T14:12:10Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819655 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |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}} m2rjzlc0ivma81q4p67wt9hht1ad3lc 6 330802 2819657 2026-07-28T14:16:46Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2819657 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260728 - 20260727) |Source={{own|Young1lim}} |Date=2026-07-28 |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}} 7z34pv8q9v6my1jtoavs88zrslpihty 10 330803 2819659 2026-07-28T14:44:41Z Earthinators 3102737 Created page with "<div style="float: left; border: 1px solid #0077b6; margin: 1px; width: 238px; font-family: sans-serif; font-size: 8pt; background: #f0f9ff; border-radius: 6px; overflow: hidden; box-shadow: 0 2px 4px rgba(0,0,0,0.1);"> <div style="float: left; width: 45px; height: 45px; background: #0077b6; text-align: center; font-size: 20pt; line-height: 45px; color: white;"> 🌍 </div> <div style="padding: 4px 8px; margin-left: 45px; color: #0f172a; line-height: 1.25em;">..." 2819659 wikitext text/x-wiki <div style="float: left; border: 1px solid #0077b6; margin: 1px; width: 238px; font-family: sans-serif; font-size: 8pt; background: #f0f9ff; border-radius: 6px; overflow: hidden; box-shadow: 0 2px 4px rgba(0,0,0,0.1);"> <div style="float: left; width: 45px; height: 45px; background: #0077b6; text-align: center; font-size: 20pt; line-height: 45px; color: white;"> 🌍 </div> <div style="padding: 4px 8px; margin-left: 45px; color: #0f172a; line-height: 1.25em;"> <b>Earthinators Circular Action Graduate</b><br/> This user completed the 5-week <b style="color:#0284c7;">Plastic Systems & Circular Solutions</b> project. </div> </div> <noinclude> [[Category:Userbox templates|Earthinators]] </noinclude> k7lfp7bso4uvb2i4lrgbbm6pl2y4upa 2819671 2819659 2026-07-28T15:12:34Z Atcovi 276019 Atcovi moved page [[Template:User Wikiversity/Earthinators]] to [[Template:User Earthinators]] without leaving a redirect: more appropriate name 2819659 wikitext text/x-wiki <div style="float: left; border: 1px solid #0077b6; margin: 1px; width: 238px; font-family: sans-serif; font-size: 8pt; background: #f0f9ff; border-radius: 6px; overflow: hidden; box-shadow: 0 2px 4px rgba(0,0,0,0.1);"> <div style="float: left; width: 45px; height: 45px; background: #0077b6; text-align: center; font-size: 20pt; line-height: 45px; color: white;"> 🌍 </div> <div style="padding: 4px 8px; margin-left: 45px; color: #0f172a; line-height: 1.25em;"> <b>Earthinators Circular Action Graduate</b><br/> This user completed the 5-week <b style="color:#0284c7;">Plastic Systems & Circular Solutions</b> project. </div> </div> <noinclude> [[Category:Userbox templates|Earthinators]] </noinclude> k7lfp7bso4uvb2i4lrgbbm6pl2y4upa 3 330805 2819676 2026-07-28T15:15:11Z Atcovi 276019 /* Earthinators: Plastic Pollution to Circular Solutions */ new section 2819676 wikitext text/x-wiki == [[Earthinators: Plastic Pollution to Circular Solutions]] == Hello Earthinators. Welcome to Wikiversity. I see you've placed your page under the "School" space, which is incorrect. [[Wikiversity:Schools|Schools]] are meant to organize resources based upon a certain field. If you want to make a specific course, it should be in the mainspace like I have demonstrated. Please make sure to read the following guidelines so your contributions can best match Wikiversity's learning missions: * [[Wikiversity:What is Wikiversity?]] * [[Wikiversity:Learning]] * [[Wikiversity:Research ethics]] * [[Wikiversity:AI]] Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:15, 28 July 2026 (UTC) skxjg3h6i31h2thwxynzwbqkgmh6jjx 2819680 2819676 2026-07-28T15:27:13Z Earthinators 3102737 /* Earthinators: Plastic Pollution to Circular Solutions */ Reply 2819680 wikitext text/x-wiki == [[Earthinators: Plastic Pollution to Circular Solutions]] == Hello Earthinators. Welcome to Wikiversity. I see you've placed your page under the "School" space, which is incorrect. [[Wikiversity:Schools|Schools]] are meant to organize resources based upon a certain field. If you want to make a specific course, it should be in the mainspace like I have demonstrated. Please make sure to read the following guidelines so your contributions can best match Wikiversity's learning missions: * [[Wikiversity:What is Wikiversity?]] * [[Wikiversity:Learning]] * [[Wikiversity:Research ethics]] * [[Wikiversity:AI]] Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:15, 28 July 2026 (UTC) :Thank You for for the warm welcome and guidance. We will make sure to read through the guidelines. Thank you for your help. [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 15:27, 28 July 2026 (UTC) gfi2n057iy05byk55akcm0smngs1icu 2819744 2819680 2026-07-29T02:46:24Z Jtneill 10242 Welcome 2819744 wikitext text/x-wiki == [[Earthinators: Plastic Pollution to Circular Solutions]] == Hello Earthinators. Welcome to Wikiversity. I see you've placed your page under the "School" space, which is incorrect. [[Wikiversity:Schools|Schools]] are meant to organize resources based upon a certain field. If you want to make a specific course, it should be in the mainspace like I have demonstrated. Please make sure to read the following guidelines so your contributions can best match Wikiversity's learning missions: * [[Wikiversity:What is Wikiversity?]] * [[Wikiversity:Learning]] * [[Wikiversity:Research ethics]] * [[Wikiversity:AI]] Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:15, 28 July 2026 (UTC) :Thank You for for the warm welcome and guidance. We will make sure to read through the guidelines. Thank you for your help. [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 15:27, 28 July 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Earthinators!'''|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]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:46, 29 July 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} gtp55mgjjhdqltdevkvwfaz7e6o1i5p 1 330806 2819689 2026-07-28T15:55:22Z Earthinators 3102737 /* Welcome and Community Advice */ new section 2819689 wikitext text/x-wiki == Welcome and Community Advice == [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 15:55, 28 July 2026 (UTC) gcij8ips5i9adwahb9gknzvioheajlg 2819690 2819689 2026-07-28T15:56:34Z Earthinators 3102737 /* Welcome and Community Advice */ 2819690 wikitext text/x-wiki == Welcome and Community Advice == Welcome to our discussion page! We are new to Wikiversity and happy to be here. We are creating a 5-week course on plastic pollution and circular solutions. We would love your help, advice, and ideas to make this course better. Thank you so much for your time and guidance! [[User:Earthinators|Earthinators]] ([[User talk:Earthinators|discuss]] • [[Special:Contributions/Earthinators|contribs]]) 15:55, 28 July 2026 (UTC) 614s2dd1jb22d23jj7ibc40go120r6c 0 330807 2819696 2026-07-28T17:36:13Z Earthinators 3102737 Created page with "{{Learning project}} {{course}} {{environmental science}} {{Under construction}} = Earthinators = Repository of Learning Resources == Courses == * [[Earthinators: Plastic Pollution to Circular Solutions|Earthinators: Plastic Pollution to Circular Solutions]] [[Category:Environmental science]] [[Category:Courses]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]]" 2819696 wikitext text/x-wiki {{Learning project}} {{course}} {{environmental science}} {{Under construction}} = Earthinators = Repository of Learning Resources == Courses == * [[Earthinators: Plastic Pollution to Circular Solutions|Earthinators: Plastic Pollution to Circular Solutions]] [[Category:Environmental science]] [[Category:Courses]] [[Category:Resources under construction]] [[Category:Environmental sciences]] [[Category:Learning projects]] ngm9fwvxd5ronrfpeqjdbackgudpbvk 0 330808 2819709 2026-07-28T20:14:59Z CommunicationLaw 2996029 new article 2819709 wikitext text/x-wiki == '''1. SOURCES OF THE LAW OF ECUADOR''' == The Ecuadorian legal framework governing communication is primarily based on the Constitution of the Republic of [[Comparative law and justice/Ecuador|Ecuador]] (2008), which recognizes freedom of expression, the right to communication, access to information, privacy, cultural diversity, and the rights of indigenous peoples. The principal statutory source is the Organic Law on Communication (Ley Orgánica de Comunicación, LOC), originally enacted in 2013 and substantially reformed in 2019. Other important sources include the Organic Law on Personal Data Protection (2021), the Organic Comprehensive Criminal Code (COIP), the Organic Law on Transparency and Access to Public Information (LOTAIP), electoral legislation regulating political advertising, consumer protection law, and telecommunications legislation. Ecuador is also bound by international treaties, particularly the American Convention on Human Rights, the International Covenant on Civil and Political Rights, and relevant decisions of the Inter-American Court of Human Rights, which possess constitutional hierarchy under Articles 11 and 417–424 of the Constitution when they recognize more favorable rights.<ref>{{Cite journal|title=Cómo deducir derechos humanos de la ley natural y de otras disciplinas {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/327|language=es-ES|doi=10.31207/ih.v12i2.327}}</ref> ---- == '''2. PRINCIPLES OF COMMUNICATION LAW & THE MEDIA''' == Ecuador recognizes communication as both a '''fundamental right''' and a '''public interest activity'''. Every person has the right to seek, receive, produce, and disseminate truthful, verified, timely, contextualized, plural, and uncensored information. The Constitution guarantees freedom of expression while promoting diversity, interculturality, multilingualism, and equal access to media. The Organic Law on Communication establishes principles including freedom of expression, editorial independence, pluralism, non-discrimination, universal access, transparency of media ownership, protection of children and adolescents, accountability of public media, and promotion of community media. The State regulates the communications sector through independent administrative authorities but may not impose prior authorization based on editorial content.<ref>{{Cite journal|title=El Derecho al Olvido en la Era Digital: Un Análisis Comparativo Entre España y Perú {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/466|language=es-ES}}</ref> ---- == '''3. CENSORSHIP & VIOLENT CONTENT''' == The Constitution expressly prohibits '''prior censorship''', except in narrowly defined circumstances involving the protection of children and adolescents in public entertainment. The Organic Law on Communication likewise forbids prior censorship by public authorities or private actors exercising dominant market power. Nevertheless, media organizations remain subject to '''subsequent liability''' for unlawful publications, including defamation, discrimination, or violations of constitutional rights. Broadcasters must classify audiovisual content according to age appropriateness and adopt measures limiting children's exposure to excessive violence, pornography, or material harmful to their development. Incitement to violence, terrorism, genocide, discrimination, or hate speech may be sanctioned under the Criminal Code. ---- == '''4. TRUTH, HONOR & TOLERANCE''' == The Constitution protects both '''freedom of expression''' and the '''rights to honor, reputation, dignity, and personal integrity'''. Individuals whose reputation has been harmed by false or inaccurate information possess the constitutional '''right of rectification and reply''', which media organizations must provide promptly under conditions established by law. Journalists are encouraged to verify facts, distinguish news from opinion, and respect professional ethics, although truthfulness is generally enforced through subsequent civil or criminal liability rather than governmental prepublication review. Ecuadorian law also prohibits discriminatory speech based on race, ethnicity, nationality, religion, sex, sexual orientation, disability, age, or other protected characteristics. Public debate receives broader constitutional protection when it concerns public officials or matters of legitimate public interest. ---- == '''5. CULTURAL AND RELIGIOUS EXPRESSIONS''' == Ecuador defines itself as an '''intercultural and plurinational State'''. Communication law protects the dissemination of cultural, linguistic, artistic, and religious expressions belonging to all communities, including indigenous peoples, Afro-Ecuadorians, and Montubio populations. Media are encouraged to promote national identity while respecting cultural diversity and religious pluralism. Public authorities may not discriminate among religious beliefs nor suppress lawful religious expression. Community media receive special legal recognition as instruments for preserving local languages, traditions, and cultural identities. Cultural programming and educational content constitute public policy objectives within Ecuador's communication system. ---- == '''6. PRIVACY & DATA PROTECTION''' == The Constitution guarantees the rights to '''privacy''', '''personal and family intimacy''', '''confidential communications''', and protection against arbitrary interference. The '''Organic Law on Personal Data Protection (2021)''' establishes comprehensive rules governing the collection, processing, storage, transfer, and deletion of personal data. Individuals possess rights of access, rectification, updating, erasure, portability, and objection to data processing. Data controllers must process information lawfully, transparently, proportionally, and for legitimate purposes. Journalistic activities enjoy certain exceptions when processing personal data serves the public interest, provided that reporting respects constitutional guarantees concerning honor, dignity, and private life. ---- == '''7. RIGHT TO BODILY, SPIRITUAL AND DIGITAL IDENTITY''' == Ecuadorian constitutional law protects the '''right to personal identity''', including one's name, image, voice, honor, reputation, nationality, ethnicity, culture, language, beliefs, and personal development. Individuals enjoy autonomy over the commercial use of their image and biometric characteristics except where legitimate public interests justify disclosure. Religious freedom protects spiritual identity and the right to profess, change, or reject any religion without coercion. The Personal Data Protection Law further safeguards digital identity by regulating biometric data, online profiling, automated decision-making, and sensitive personal information. Unauthorized identity theft, digital impersonation, or misuse of biometric information may give rise to civil, administrative, or criminal liability. ---- == '''8. RIGHT TO REJECT INFORMATION, CLOTHING & HUMAN EXHIBITIONS''' == Ecuadorian constitutional principles recognize human dignity as the foundation of all rights. Individuals generally possess the freedom to '''accept or reject unwanted communications''', commercial advertising, intrusive publicity, or unauthorized use of their personal image. Personal autonomy also protects decisions regarding clothing, appearance, religious attire, and cultural dress, subject only to reasonable legal limitations based on public safety, health, or institutional requirements. No person may be publicly exhibited, photographed, filmed, or commercially exploited in a manner that violates dignity, privacy, or personal integrity without lawful justification or valid consent. Special protection is afforded to children, victims of crime, persons deprived of liberty, and other vulnerable groups against degrading or sensationalistic media exposure. Digital dissemination of intimate images without consent may result in civil damages and criminal sanctions under Ecuadorian law. ts9udddlobvcjk4ngy4tjip3ljjed2h 2819710 2819709 2026-07-28T20:16:24Z CommunicationLaw 2996029 include references 2819710 wikitext text/x-wiki == '''1. SOURCES OF THE LAW OF ECUADOR''' == The Ecuadorian legal framework governing communication is primarily based on the Constitution of the Republic of [[Comparative law and justice/Ecuador|Ecuador]] (2008), which recognizes freedom of expression, the right to communication, access to information, privacy, cultural diversity, and the rights of indigenous peoples. The principal statutory source is the Organic Law on Communication (Ley Orgánica de Comunicación, LOC), originally enacted in 2013 and substantially reformed in 2019. Other important sources include the Organic Law on Personal Data Protection (2021), the Organic Comprehensive Criminal Code (COIP), the Organic Law on Transparency and Access to Public Information (LOTAIP), electoral legislation regulating political advertising, consumer protection law, and telecommunications legislation. Ecuador is also bound by international treaties, particularly the American Convention on Human Rights, the International Covenant on Civil and Political Rights, and relevant decisions of the Inter-American Court of Human Rights, which possess constitutional hierarchy under Articles 11 and 417–424 of the Constitution when they recognize more favorable rights.<ref>{{Cite journal|title=Cómo deducir derechos humanos de la ley natural y de otras disciplinas {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/327|language=es-ES|doi=10.31207/ih.v12i2.327}}</ref> ---- == '''2. PRINCIPLES OF COMMUNICATION LAW & THE MEDIA''' == Ecuador recognizes communication as both a '''fundamental right''' and a '''public interest activity'''. Every person has the right to seek, receive, produce, and disseminate truthful, verified, timely, contextualized, plural, and uncensored information. The Constitution guarantees freedom of expression while promoting diversity, interculturality, multilingualism, and equal access to media. The Organic Law on Communication establishes principles including freedom of expression, editorial independence, pluralism, non-discrimination, universal access, transparency of media ownership, protection of children and adolescents, accountability of public media, and promotion of community media. The State regulates the communications sector through independent administrative authorities but may not impose prior authorization based on editorial content.<ref>{{Cite journal|title=El Derecho al Olvido en la Era Digital: Un Análisis Comparativo Entre España y Perú {{!}} Ius Humani. Revista de Derecho|url=https://www.iushumani.org/index.php/iushumani/article/view/466|language=es-ES}}</ref> ---- == '''3. CENSORSHIP & VIOLENT CONTENT''' == The Constitution expressly prohibits '''prior censorship''', except in narrowly defined circumstances involving the protection of children and adolescents in public entertainment. The Organic Law on Communication likewise forbids prior censorship by public authorities or private actors exercising dominant market power. Nevertheless, media organizations remain subject to '''subsequent liability''' for unlawful publications, including defamation, discrimination, or violations of constitutional rights. Broadcasters must classify audiovisual content according to age appropriateness and adopt measures limiting children's exposure to excessive violence, pornography, or material harmful to their development. Incitement to violence, terrorism, genocide, discrimination, or hate speech may be sanctioned under the Criminal Code. ---- == '''4. TRUTH, HONOR & TOLERANCE''' == The Constitution protects both '''freedom of expression''' and the '''rights to honor, reputation, dignity, and personal integrity'''. Individuals whose reputation has been harmed by false or inaccurate information possess the constitutional '''right of rectification and reply''', which media organizations must provide promptly under conditions established by law. Journalists are encouraged to verify facts, distinguish news from opinion, and respect professional ethics, although truthfulness is generally enforced through subsequent civil or criminal liability rather than governmental prepublication review. Ecuadorian law also prohibits discriminatory speech based on race, ethnicity, nationality, religion, sex, sexual orientation, disability, age, or other protected characteristics. Public debate receives broader constitutional protection when it concerns public officials or matters of legitimate public interest. ---- == '''5. CULTURAL AND RELIGIOUS EXPRESSIONS''' == Ecuador defines itself as an '''intercultural and plurinational State'''. Communication law protects the dissemination of cultural, linguistic, artistic, and religious expressions belonging to all communities, including indigenous peoples, Afro-Ecuadorians, and Montubio populations. Media are encouraged to promote national identity while respecting cultural diversity and religious pluralism. Public authorities may not discriminate among religious beliefs nor suppress lawful religious expression. Community media receive special legal recognition as instruments for preserving local languages, traditions, and cultural identities. Cultural programming and educational content constitute public policy objectives within Ecuador's communication system. ---- == '''6. PRIVACY & DATA PROTECTION''' == The Constitution guarantees the rights to '''privacy''', '''personal and family intimacy''', '''confidential communications''', and protection against arbitrary interference. The '''Organic Law on Personal Data Protection (2021)''' establishes comprehensive rules governing the collection, processing, storage, transfer, and deletion of personal data. Individuals possess rights of access, rectification, updating, erasure, portability, and objection to data processing. Data controllers must process information lawfully, transparently, proportionally, and for legitimate purposes. Journalistic activities enjoy certain exceptions when processing personal data serves the public interest, provided that reporting respects constitutional guarantees concerning honor, dignity, and private life. ---- == '''7. RIGHT TO BODILY, SPIRITUAL AND DIGITAL IDENTITY''' == Ecuadorian constitutional law protects the '''right to personal identity''', including one's name, image, voice, honor, reputation, nationality, ethnicity, culture, language, beliefs, and personal development. Individuals enjoy autonomy over the commercial use of their image and biometric characteristics except where legitimate public interests justify disclosure. Religious freedom protects spiritual identity and the right to profess, change, or reject any religion without coercion. The Personal Data Protection Law further safeguards digital identity by regulating biometric data, online profiling, automated decision-making, and sensitive personal information. Unauthorized identity theft, digital impersonation, or misuse of biometric information may give rise to civil, administrative, or criminal liability. ---- == '''8. RIGHT TO REJECT INFORMATION, CLOTHING & HUMAN EXHIBITIONS''' == Ecuadorian constitutional principles recognize human dignity as the foundation of all rights. Individuals generally possess the freedom to '''accept or reject unwanted communications''', commercial advertising, intrusive publicity, or unauthorized use of their personal image. Personal autonomy also protects decisions regarding clothing, appearance, religious attire, and cultural dress, subject only to reasonable legal limitations based on public safety, health, or institutional requirements. No person may be publicly exhibited, photographed, filmed, or commercially exploited in a manner that violates dignity, privacy, or personal integrity without lawful justification or valid consent. Special protection is afforded to children, victims of crime, persons deprived of liberty, and other vulnerable groups against degrading or sensationalistic media exposure. Digital dissemination of intimate images without consent may result in civil damages and criminal sanctions under Ecuadorian law. == References == c8dlg6sgldepn3abhgu7qg0upvuvpqi 3 330809 2819715 2026-07-28T22:07:23Z Jtneill 10242 Welcome 2819715 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], U3280251!'''|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]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:07, 28 July 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 3um1hbq4m886w1atwpfakbfxjwgmc4h 3 330810 2819716 2026-07-28T22:08:11Z Jtneill 10242 Welcome 2819716 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], U3285438!'''|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]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:08, 28 July 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} crfjqjhtlxyp4rewribgkwsc3ohvozw 118 330811 2819717 2026-07-28T23:49:41Z Brunnaiz 2943152 Copying from [[Introduction to Swedish/Alphabet]]. My goal is to create a basic Catalan course 2819717 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Sv-alphabet.ogg|The Swedish alphabet.|right|thumb]] The Swedish alphabet contains 29 letters: Aa Bb Cc Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz Åå Ää Öö The last three letters, Å, Ä, and Ö, are regarded as distinct letters, separate from A and O. Whereas in English you might use the term "A - Z", in Swedish it would be "A - Ö". Unlike in English, Y is always a vowel, except in the case of some loanwords. W has not been included in the Swedish alphabet for a long time, but since 2006 the Swedish Academy considers W as a separate letter. However, this is not fully accepted in dictionaries and other instances yet. ==The Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==The Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are when multiple letters are used in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |dj, gj, hj, lj |y |- |sj, skj, stj |Sounds like ''wh'' in old-fashioned southern US accents |- |kj, tj |sh |- |hv |v |- |ng |ng |- |gn |ng (''gn'' is not a digraph if at the beginning of a word, where it would be pronounced like ''a'''gn'''ostic'') |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' [[Category:Catalan]] [[Category:Alphabets]] 0ymzkhi1j6b186l4w63yq121ou3s0fg 2819718 2819717 2026-07-28T23:53:44Z Brunnaiz 2943152 2819718 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The last three letters, Å, Ä, and Ö, are regarded as distinct letters, separate from A and O. Whereas in English you might use the term "A - Z", in Swedish it would be "A - Ö". Unlike in English, Y is always a vowel, except in the case of some loanwords. W has not been included in the Swedish alphabet for a long time, but since 2006 the Swedish Academy considers W as a separate letter. However, this is not fully accepted in dictionaries and other instances yet. ==The Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==The Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are when multiple letters are used in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |dj, gj, hj, lj |y |- |sj, skj, stj |Sounds like ''wh'' in old-fashioned southern US accents |- |kj, tj |sh |- |hv |v |- |ng |ng |- |gn |ng (''gn'' is not a digraph if at the beginning of a word, where it would be pronounced like ''a'''gn'''ostic'') |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' [[Category:Catalan]] [[Category:Alphabets]] b4zbznlfco9sbtxwmc3bf6gufdz883x 2819721 2819718 2026-07-29T00:03:08Z Brunnaiz 2943152 +information coming from [[:w:Cedilla]] 2819721 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The only letter that doesn't exist in the [[English alphabet]] is the [[w:Cedilla|cedilla]] (ç), which is also used in other Romance languages, such as French and Portuguese. For example, English rarely features it in loanwords from French and Portuguese such as ''façade'', ''limaçon'' and ''cachaça'' (often typed ''facade'', ''limacon'' and ''cachaca'', respectively, because of lack of ç keys on English-language keyboards). ==The Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==The Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are when multiple letters are used in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |dj, gj, hj, lj |y |- |sj, skj, stj |Sounds like ''wh'' in old-fashioned southern US accents |- |kj, tj |sh |- |hv |v |- |ng |ng |- |gn |ng (''gn'' is not a digraph if at the beginning of a word, where it would be pronounced like ''a'''gn'''ostic'') |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' [[Category:Catalan]] [[Category:Alphabets]] d9d73y7fs41nvz9zpr556lhl91dxw3g 2819722 2819721 2026-07-29T00:04:15Z Brunnaiz 2943152 +information coming from [[:w:Cedilla]] 2819722 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The only letter that doesn't exist in the [[English alphabet]] is the [[w:Cedilla|cedilla]] (ç), which in Catalan is called ''ce trencada'' (i.e. "broken C"). It's also used in other Romance languages, such as French and Portuguese. For example, English rarely features it in loanwords from French and Portuguese such as ''façade'', ''limaçon'' and ''cachaça'' (often typed ''facade'', ''limacon'' and ''cachaca'', respectively, because of lack of ç keys on English-language keyboards). ==The Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==The Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are when multiple letters are used in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |dj, gj, hj, lj |y |- |sj, skj, stj |Sounds like ''wh'' in old-fashioned southern US accents |- |kj, tj |sh |- |hv |v |- |ng |ng |- |gn |ng (''gn'' is not a digraph if at the beginning of a word, where it would be pronounced like ''a'''gn'''ostic'') |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' [[Category:Catalan]] [[Category:Alphabets]] dg6ic8sj6oa97plbxs3s6a3j86mlhne 2819723 2819722 2026-07-29T00:07:15Z Brunnaiz 2943152 +information coming from [[:w:Cedilla]] 2819723 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The only letter that doesn't exist in the [[English alphabet]] is the [[w:Cedilla|cedilla]] (ç), which in Catalan is called ''ce trencada'' (i.e. "broken C").<ref>{{Cite book|title=[[:w:IEC Dictionary of the Catalan Language|Diccionari de la llengua catalana]]|publisher=[[:w:Institute for Catalan Studies|Institut d'Estudis Catalans]]|chapter=ce|access-date=29 July 2026|chapter-url=https://dlc.iec.cat/Results?DecEntradaText=ce&AllInfoMorf=False&OperEntrada=0&OperDef=0&OperEx=0&OperSubEntrada=0&OperAreaTematica=0&InfoMorfType=0&OperCatGram=False&AccentSen=False&CurrentPage=0&refineSearch=0&Actualitzacions=False}}</ref> It's also used in other Romance languages, such as French and Portuguese. For example, English rarely features it in loanwords from French and Portuguese such as ''façade'', ''limaçon'' and ''cachaça'' (often typed ''facade'', ''limacon'' and ''cachaca'', respectively, because of lack of ç keys on English-language keyboards). ==The Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==The Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are when multiple letters are used in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |dj, gj, hj, lj |y |- |sj, skj, stj |Sounds like ''wh'' in old-fashioned southern US accents |- |kj, tj |sh |- |hv |v |- |ng |ng |- |gn |ng (''gn'' is not a digraph if at the beginning of a word, where it would be pronounced like ''a'''gn'''ostic'') |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' [[Category:Catalan]] [[Category:Alphabets]] rjgypyn1aoj1em5cji9ee2puzyer3kd 2819724 2819723 2026-07-29T00:09:09Z Brunnaiz 2943152 +{{References}} 2819724 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The only letter that doesn't exist in the [[English alphabet]] is the [[w:Cedilla|cedilla]] (ç), which in Catalan is called ''ce trencada'' (i.e. "broken C").<ref>{{Cite book|title=[[:w:IEC Dictionary of the Catalan Language|Diccionari de la llengua catalana]]|publisher=[[:w:Institute for Catalan Studies|Institut d'Estudis Catalans]]|chapter=ce|access-date=29 July 2026|chapter-url=https://dlc.iec.cat/Results?DecEntradaText=ce&AllInfoMorf=False&OperEntrada=0&OperDef=0&OperEx=0&OperSubEntrada=0&OperAreaTematica=0&InfoMorfType=0&OperCatGram=False&AccentSen=False&CurrentPage=0&refineSearch=0&Actualitzacions=False}}</ref> It's also used in other Romance languages, such as French and Portuguese. For example, English rarely features it in loanwords from French and Portuguese such as ''façade'', ''limaçon'' and ''cachaça'' (often typed ''facade'', ''limacon'' and ''cachaca'', respectively, because of lack of ç keys on English-language keyboards). ==The Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==The Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are when multiple letters are used in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |dj, gj, hj, lj |y |- |sj, skj, stj |Sounds like ''wh'' in old-fashioned southern US accents |- |kj, tj |sh |- |hv |v |- |ng |ng |- |gn |ng (''gn'' is not a digraph if at the beginning of a word, where it would be pronounced like ''a'''gn'''ostic'') |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' ==References== {{References}} [[Category:Catalan]] [[Category:Alphabets]] i7y974z80nmcf2gduzapocu88ehg331 2819725 2819724 2026-07-29T00:11:10Z Brunnaiz 2943152 +information 2819725 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The only letter that doesn't exist in the [[English alphabet]] is the [[w:Cedilla|cedilla]] (ç), which in Catalan is called ''ce trencada'' (i.e. "broken C").<ref>{{Cite book|title=[[:w:IEC Dictionary of the Catalan Language|Diccionari de la llengua catalana]]|publisher=[[:w:Institute for Catalan Studies|Institut d'Estudis Catalans]]|chapter=ce|access-date=29 July 2026|chapter-url=https://dlc.iec.cat/Results?DecEntradaText=ce&AllInfoMorf=False&OperEntrada=0&OperDef=0&OperEx=0&OperSubEntrada=0&OperAreaTematica=0&InfoMorfType=0&OperCatGram=False&AccentSen=False&CurrentPage=0&refineSearch=0&Actualitzacions=False}}</ref> It's also used in other Romance languages, such as French and Portuguese. For example, English rarely features it in loanwords from French and Portuguese such as ''façade'', ''limaçon'' and ''cachaça'' (often typed ''facade'', ''limacon'' and ''cachaca'', respectively, because of lack of ç keys on English-language keyboards). ==Vowels== Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Long Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Short Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are multiple letters which are used together in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |gu |g |- |qu |c |- |ny | |- |tx |ch |- |ix |sh |- |ll | |- |l·l |ll |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' ==References== {{References}} [[Category:Catalan]] [[Category:Alphabets]] 1rm1td0lq7rvk56v7djs8x6si84h0c5 2819726 2819725 2026-07-29T00:14:27Z Brunnaiz 2943152 /* Vowels */ 2819726 wikitext text/x-wiki <!--A pronunciation guide has been added, but may be to complicated. If someone could organize it so that it lists the pronunciations by letter it would be greatly appreciated.--> {{:Introduction to Swedish/Navbar}} ---- [[File:Ca-alphabet.ogg|The Catalan alphabet.|right|thumb]] The Catalan alphabet contains 27 letters: Aa Bb Cc Çç Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The only letter that doesn't exist in the [[English alphabet]] is the [[w:Cedilla|cedilla]] (ç), which in Catalan is called ''ce trencada'' (i.e. "broken C").<ref>{{Cite book|title=[[:w:IEC Dictionary of the Catalan Language|Diccionari de la llengua catalana]]|publisher=[[:w:Institute for Catalan Studies|Institut d'Estudis Catalans]]|chapter=ce|access-date=29 July 2026|chapter-url=https://dlc.iec.cat/Results?DecEntradaText=ce&AllInfoMorf=False&OperEntrada=0&OperDef=0&OperEx=0&OperSubEntrada=0&OperAreaTematica=0&InfoMorfType=0&OperCatGram=False&AccentSen=False&CurrentPage=0&refineSearch=0&Actualitzacions=False}}</ref> It's also used in other Romance languages, such as French and Portuguese. For example, English rarely features it in loanwords from French and Portuguese such as ''façade'', ''limaçon'' and ''cachaça'' (often typed ''facade'', ''limacon'' and ''cachaca'', respectively, because of lack of ç keys on English-language keyboards). ==Vowels== There are 8 vowels in Catalan and other semivowels. Vowels are divided into short and long vowels, similarly to English. Just as in English, the short and long versions of vowels are pronounced differently. When reading new words you should pay attention to whether the vowels are short or long. The basic rule is that a vowel is short if it is followed by more than one consonant, similar to English. Much like English, though, there will be exceptions to the rule. ===Vowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|ɑː}} || {{Audio|sv-mat.ogg|listen}} ''mat'', {{IPA|/mɑːt/}}, "food" |''ar'' in ''start'' (British English) |- |E | {{IPA|eː}} || {{Audio|sv-hel.ogg|listen}} ''hel'', {{IPA|/heːl/}}, "whole" |Like ''ea'' in ''yeah,'' but with wide lips |- |E | | | |- |I | {{IPA|iː}} || {{Audio|sv-sil.ogg|listen}} ''sil'', {{IPA|/siːl/}}, "sieve" |''ee'' in ''sheet'' (British English) |- |O | {{IPA|uː}} || {{Audio|sv-bot.ogg|listen}} ''bot'', {{IPA|/buːt/}}, "penance" |''oo'' in ''pool'' (British English) |- |U | {{IPA|ʉ̟ː}} || {{Audio|sv-ful.ogg|listen}} ''ful'', {{IPA|/fʉ̟ːl/}}, "ugly" |''oo'' in ''pool'' (Australian English) |- |Y | {{IPA|yː}} || {{Audio|sv-syl.ogg|listen}} ''syl'', {{IPA|/syːl/}}, "[[wikt:awl|awl]]" |Like ''ee'' in ''sheet'', but with circular lips (British English) |- |Å | {{IPA|oː}} || {{Audio|sv-mål.ogg|listen}} ''mål'', {{IPA|/moːl/}}, "goal" |''aw'' in ''law'' (British English) |- |Ä | {{IPA|ɛː}} || {{Audio|sv-häl.ogg|listen}} ''häl'', {{IPA|/hɛːl/}}, "heel" |''ea'' in ''yeah'' |- |Ö | {{IPA|øː}} || {{Audio|sv-nöt.ogg|listen}} ''nöt,'' {{IPA|/nøːt/}}, "nut" |''or'' in ''work'' (British English) |} ===Semivowels=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- !Letter ! align="center" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="center" | Pronunciation sample, phonemic transcription and translation !English equivalent |- |A | {{IPA|a}} || {{Audio|sv-matt.ogg|listen}} ''matt'', {{IPA|/mat/}}, "listless; matte" |None |- |E, Ä | {{IPA|ɛ}}|| {{Audio|sv-häll.ogg|listen}} ''häll'', {{IPA|/hɛl/}}, "flat rock" |''e'' in ''bed'' |- |I | {{IPA|ɪ}}|| {{Audio|sv-sill.ogg|listen}} ''sill'', {{IPA|/sɪl/}}, "herring" |''i'' in ''sit'' |- |O | {{IPA|ɔ}} || {{Audio|sv-moll.ogg|listen}} ''moll'', {{IPA|/mɔl/}}, "minor" (music) |''o'' in ''toll'' (British English) |- |O | {{IPA|ʊ}} || {{Audio|sv-bott.ogg|listen}} ''bott'', {{IPA|/bʊt/}}, "lived" ([[w:perfect tense|perfect tense]]) |''oo'' in ''look'' |- |U | {{IPA|ɵ}} || {{Audio|sv-full.ogg|listen}} ''full'', {{IPA|/fɵl/}}, "full" |Similar to ''Nevad'''<u>a</u>''''' |- |Y | {{IPA|ʏ}} || {{Audio|sv-syll.ogg|listen}} ''syll'', {{IPA|/sʏl/}}, "[[w:railroad tie|sleeper]]" (railroad) |Like ''i'' in ''sit'', but with circular lips |- |Ö | {{IPA|œ}} || {{Audio|sv-nött.ogg|listen}} ''nött'', {{IPA|/nœt/}}, "worn" |Like ''or'' in ''work,'' but shorter |} === Influence of R === ''e'' and ''ä'' are pronounced like the ''a'' in English ''tr'''a'''p'' if before an r. This means ''ärt'' is pronounced similarly to English ''at''. ==Consonants== The consonants are mostly similar to English. Most Swedes do not roll the ''r'', which makes Swedish more comfortable for English speakers compared to other European languages. The letters ''z'', ''q'', and ''w'' are very uncommon but are used in family and company names. The most obvious differences for English speakers will be that ''y'' is never a consonant, and ''j'' is pronounced like English's consonantal ''y''. '''Long Consonants''' As in English, and as demonstrated in the previous section, consonants may be doubled. However, if this happens in the middle of the word (such as ''erkänna''), then the doubled consonant is pronounced with double the length. This is similar to how the word ''night-time'' in English is spoken with a doubly long ''t''. '''Initial ''g'' and ''k'' ''' There is a similar concept to how ''g'' is pronounced differently in the English words ''general'' and ''game'', only with slightly different rules and a different pronunciation. The letter ''k'' also changes pronunciation in Swedish based on the following vowel. Generally, if the first letter in a word is ''g'' and if the next letter is an ''e'', ''i'', ''ä'', ''ö'', or ''y'', then the ''g'' is pronounced as an English ''y''.<br> Generally, if the first letter in a word is ''k'' and if the next letter is an e, ''i'', ''ä'', ''ö'', or ''y'', then the ''k'' is pronounced as an English ''sh''. '''Digraphs''' Digraphs are multiple letters which are used together in writing to represent a single spoken sound. This also occurs in English (consider ''sh''). {| class="wikitable" |+ !Digraph !English equivalent |- |gu |g |- |qu |c |- |ny | |- |tx |ch |- |ix |sh |- |ll | |- |l·l |ll |} ===Plosives=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|p}} || {{Audio|sv-pol.ogg|listen}} ''pol'', {{IPA|/puːl/}}, "pole" (of axis) |- | {{IPA|b}} || {{Audio|sv-bok.ogg|listen}} ''bok'', {{IPA|/buːk/}}, "book" |- | {{IPA|t}} || {{Audio|sv-tok.ogg|listen}} ''tok'', {{IPA|/tuːk/}}, "fool" |- | {{IPA|d}} || {{Audio|sv-dop.ogg|listen}} ''dop'', {{IPA|/duːp/}}, "christening" |- | {{IPA|k}} || {{Audio|sv-kon.ogg|listen}} ''kon'', {{IPA|/kuːn/}}, "cone" |- | {{IPA|ɡ}} || {{Audio|sv-god.ogg|listen}} ''god'', {{IPA|/ɡuːd/}}, "good" |} ===Fricative=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|f}} || {{Audio|sv-fot.ogg|listen}} ''fot'', {{IPA|/fuːt/}}, "foot" |- | {{IPA|v}} || {{Audio|Sv-våt.ogg|listen}} ''våt'', {{IPA|/voːt/}}, "wet" |- | {{IPA|s}} || {{Audio|sv-sot.ogg|listen}} ''sot'', {{IPA|/suːt/}}, "soot" |- | {{IPA|ɧ}} || {{Audio|sv-sjok.ogg|listen}} ''sjok'', {{IPA|/ɧuːk/}}, "chunk" |- | {{IPA|ɕ}} || {{Audio|sv-kjol.ogg|listen}} ''kjol'', {{IPA|/ɕuːl/}}, "skirt" |- | {{IPA|j}} || {{Audio|sv-jord.ogg|listen}} ''jord'', {{IPA|/juːrd/}}, "soil, earth" |- | {{IPA|h}} || {{Audio|sv-hot.ogg|listen}} ''hot'', {{IPA|/huːt/}}, "threat" |} ===/r/ and Retroflex Assimilations=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|r}} || {{Audio|sv-rov.ogg|listen}} ''rov'', {{IPA|/ruːv/}}, "prey; loot" |- |} ===Laterals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|l}} || {{Audio|sv-lov.ogg|listen}} ''lov'', {{IPA|/luːv/}}, "[[w:Tack (sailing)|tack (sailing maneuver)]]" |- |} ===Nasals=== {| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-family: Arial Unicode MS, Lucida Sans Unicode" |- ! align="left" | [[w:phoneme|Phoneme]]<br>([[w:International Phonetic Alphabet|IPA]]) ! align="left" | Pronunciation sample, phonemic transcription and translation |- | {{IPA|m}} || {{Audio|sv-mod.ogg|listen}} ''mod'', {{IPA|/muːd/}}, "courage" |- | {{IPA|n}} || {{Audio|sv-nod.ogg|listen}} ''nod'', {{IPA|/nuːd/}}, "node" |- | {{IPA|ŋ}} || {{Audio|sv-lång.ogg|listen}} ''lång'', {{IPA|/lɔŋ/}}, "long" |} ==Stress== Normally the stress is put on the first vowel. The tonality/pitch is more important in some Swedish dialects. ==Tonality== Both ''nita'' and ''niten'' have the stress on the vowel ''i''. However, in some accents the tonality/pitch differs. This is often a situational difference that varies from one word to the next. For more information on tone you can try the Wikipedia article on [[w:tone (linguistics)|tone]]. ==Typing Special Characters== [[Image:Lange Jan.jpg|The light house "Långe Jan" at the south end of Öland island.|thumb]] '''On a PC:''' Alt+0229 = å<br> Alt+0197 = Å<br> Alt+0228 = ä<br> Alt+0196 = Ä<br> Alt+0246 = ö<br> Alt+0214 = Ö '''On a Mac:''' Option+a = å<br> Option+A = Å<br> Option+u to get ¨ then type a = ä<br> Option+u to get ¨ then type A = Ä<br> Option+u to get ¨ then type o = ö<br> Option+u to get ¨ then type O = Ö<br> If you don't manage to get these characters, the standard way is to substitute '''å''' with '''aa''', '''ä''' with '''ae''', and '''ö''' with '''oe'''. ==More Information on Pronunciation== For more detailed information on Swedish pronunciations read the Wikipedia article on [[w:swedish phonology|Swedish phonology]]. ==Exercise== Pronounce the following Swedish words. 1. ''skal, sno, tur, kår, ven, mil, syl, när, bör'' 2. ''stall, stopp, lupp, sådd, ett, stins, skyll, ärr, börs'' 3. ''vara, bliva, heta, kallas, äta, dricka, festa, leka'' 4. ''bilen, vägen, äpplet, trädet, smaken, tiken'' 5. ''kal, kotte, kul, kål, gam, gott, gurka, gås'' 6. ''ge, gick, gylf, Gävle, gök, kedja, kil, kyl, käk, kök'' ==References== {{References}} [[Category:Catalan]] [[Category:Alphabets]] acc4y2a1q11f3izfyezvfg0395edrgx 3 330812 2819720 2026-07-28T23:55:45Z PieWriter 3039865 Created page with "{{subst:welcome}}" 2819720 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Brunnaiz!'''|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:PieWriter|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:55, 28 July 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 2gbsci0l5y11thq6zxvhvyh5c6cjs9n